All you can do is act like typical parents - and in many ways you are, but its always worth bearing in mind that your kids just maybe aren't going to be typical kids! Adopted kids often suffer from social thinning, which also leaves them vulnerable to loneliness and mental health issues later on in life. So you can be proactive and support your child in building lots of varying social networks- scouts, guides, swim team, school, church, gardening club, whatever. It also means you are not as dependant on one should things go wrong.
In regards to dealing with the bullying itself, you have a lot to consider. I worked in a secondary school, and there is no one size fits all approach, because it depends on the social standing of each child involved. I have a few thoughts that are based entirely on my own personal views.
What I should officially say is:
Obviously tell your child's teacher and follow the policy etc.
However....
Bullying is rarely simple, and also rarely solved by the application of a policy, if at all. Schools have limited resources. There is no policy in the world that can accomodate all the nuances of bullying, and so relying on what "should" happen is rarely going to improve the situation. What I'm going to suggest is what I would actually be doing or considering to genuinely support my kid in a typical case of bullying. It focuses less on dealing with the bully and more about my child- what do I need to do to prioritise my child's wellbeing and behaviour, rather than change the bully. The bully is unlikely to change at a speed that is necessary to prevent damage to my child, so the actions I take need to be about me influencing and taking control over what I can- i.e my child's actions and drawing on the relationships with staff that I can utilise to ensure their safety.
There is a caveat that any advice below this depends on the temperament of your child. My kids both dish it with varying degrees of severity, so the responses will be different to if we had a nice quiet gentle child whose first response was to weep and not to just deck someone immediately when they said something dumb. (The woes of parenting children who are so vastly different to you.... the chances of my two suffering anything in silence is slim to none.)
Anyway, here's what I would do.
Information gathering:
My question when they tell me someone was mean is always what was happening just before that? Getting them to work out what the trigger was can either be self reflection for them, or can help you work out why and when it happens. It can help a teacher know what to look out for as well. Think about any previous incidents and work put whether your child is (rightly or wrongly) linking them all together. Some kids just don't get on- it isn't bullying, they just don't like one another. Make sure you are accurately recording what your child says happened with dates and times, and then remember to tell the school. They can only deal with what they know. It's infuriating to be told something has gone on for ages when the school has had no idea but was expected to sort it out immediately. Be polite and non judgemental of the other child where possible.
Work out where your child is in the social standing.
Be open to the idea that it is entirely possible that your child (due to attachment issues, developmental difficulties etc) is batting above their social capital. Many a time, particularly in year 3/4 and year 7/8 it seems to me, you see kids trying to befriend the most popular kid in the class when (to be brutally honest) they are just not cool or mature enough to have an equitable relationship with them. If your kid is immature, or socially awkward, it is often best to try and redirect them if they are being continually rejected by a specific person. It can feel like bullying, but a skilled teacher or pastoral lead will be able to try and engineer some socially appropriate relationships where the child won't be constantly rejected. There is only so much that an 8 year old can do to reject someone nicely. That being said, there is a social power differential to consider- if the popular kid is ALWAYS mean, or a kid slightly lower in the pecking order isnbeig mean to them constantly to try and get laughs or recognition, that needs to be recognised too. It's like the saying that true humour punches up- the popular kid has a responsibility to be kind in their rejection, and the victim should not be used as social leverage. However, nobody owes anyone else friendship, so that is an important distinction to make.
You also need to work put the root cause. In secondary, things like throwing paper at a kid who they know will kick off in a supply lesson and then get kicked out to derail the whole lesson is common and allows a bully to get social kudos for getting the class out of lesson, for example. Kids will say all sorts of horrible stuff to someone to get a rise out of them and derail a lesson, and it is important to recognise that their ultimate goal was to get out of history, not necessarily cause lasting psychological harm to your child, although that may be the effect. There's work to be done on a lot of levels there, and I would be working with the teacher to yes, acknowledge what my kid had done (flipped the table, shouted and screamed etc) and how this could be best dealt with next time (support for the supply, an exit card for my kid, conversation with my kid saying that they were believed but also that it wasn't ok, getting them to recognise that their reaction exacerbated the situation so using their calming strategies will mean they can deal with it in a way that is more effective and doesnt get them into trouble) AND looking for the staff to speak with the bully and work out what could be done about the teachers behaviour management plan. Why was any of this happening in the lesson in the first place? Kids will be kids, nobody likes a supply, but it's not acceptable to do that.
It may well also be worth considering that the kid might have been told not to socialise with yours outside of class. It's not acceptable, but a harsh reality. Some kids won't come to the birthday party and won't invite yours because of their (often extreme religious) beliefs. It's a shame but fighting that is very tricky, and can put younger kids in a difficult situation. Teenagers are usually a bit easier in this way because they've realised their parents don't see them at schooland will develop their own ideas, but then again, they can also become fixated on their own identity that means they want to be seen doing things the way their family wants them to. Depending on the age of your child and their capacity to understand, you might have to explain that to them.
Determine the child's desires
I try to be non judgemental, and not to jump in saying nobody should hurt my darling boy and he needs to stay well away from them, because it is worth the time to acknowledge that there is possibly a relationship that they ultimately want to protect (lots of school aged bullying is prior friends falling out) so determining whether they want to continue the relationship in some form (maybe they are all part of the same friendhsip group) and therefore need strategies about how to manage a toxic friend, or whether they need adult intervention to genuinely stop or minimise all contact between them is key. Do they have the words to accurately describe what is happening? Is it a falling out or genuine bullying? Maybe for teenagers it is banter. And there is no point in saying they shouldn't banter, because they can will and do- so how should your kid manage it? Do they have a line they don't want crossed, and how could they effectively communicate this? The mayor of London did an ad about tackling street harassment that I found helpful to frame conversations that weren't about specific incidents but gave some general ideas about how to respond in a way that showed disapproval without disrupting the relationship.
Its also worth considering do they actually want you to intervene? Or do they just want to vent to you? I have had many a weeping teenage girl in my office devastated that her dad came up the shool to shout and threaten some of the boys who she said were bullying her, and it made the situation much worse. What they usually wanted was to be able to tell their parents and have them comfort her, and maybe fix whatever the problem was- more deodorant, help to lose weight, sympathy over not having a boyfriend, a better social media picture etc. What are they actually upset about? Nobody gets upset about something that is completely untrue, often it is something that the kids is worried about deep down anyway. It can also help you work out a reasonable solution with the school. It is rarely reasonable to expect the bully to be flayed alive and hung up by their toenails (no matter how much you would like that) , but if you can work out what the issue is, you can give some realistic and achievable solutions that won't make your kid a social pariah. Things that have worked have been things like discreetly moving seating plans so that your child is away from the bully, doing whole year group or whole school assemblies, asking the school to mark lgbt history month, providing a place at lunch for your child to go to, restorative justice sessions with a review date, asking all parties to hand their phones in at the start of the day, a designated person to tell if something happens, specific conversations to be had with a child about any misconceptions that have arisen etc.
Also consider the stuff that you can do with your kid to improve their resilience once you find the root of why this makes them feel vulnerable. Maybe they are worried about their weight, could you help them with that? Maybe they are worried they don't have a boyfriend or girlfriend, can you commiserate or find other social opportunities with them?
If the bullying is based in homophobia or transphobia, it is unlikely that what has been said will be an isolated incident, so depending on the comment you can support your child's feelings about it before dealing with the incident. If it was a dig at you, it can help to acknowledge that the feelings around that for your child are complicated and possibly difficult. Did they wish they had a snappy comeback and are replaying it? Worried that the bully was right? Ashamed that they did actually feel embarrassed about you? Worried you will be offended by what their friends say and not allow them to hang out with them? Are they consequently worried about you rejecting them for feeling that way but also worried about their friends rejecting them? Maybe they assessed the situation and didnt feel safe enough to challenge it? All those feelings and actions are valid, but might make them feel conflicted. To me it is important to explicitly state to them that the opinion of other children about you doesn't matter to you because you are a secure adult, your main concern is their wellbeing. Obviously as grown ups you can know that people like that aren't their friends really, but to a kid it feels vitally important, and they can feel really really split about where their loyalties lie. I would reassure them regularly that our love is never in question and that whilst we are proud of who we are, we also want to make them happy and safe and comfortable. We want to help them decide how to get what they want out of the relationship with this person, and support them doing it.
It is also worth being clear about what youn can and cannot offer to help them. For example, some people don't like the idea of rejecting their identity, but I regularly ask my kids to dress smartly and sit quietly at family events when they clearly identify as hurricanes in human form, so if in their teenage years that means they want me to pick them up in a car that doesn't have a pride sticker on or wearing one of the many pride tshirts that I've accumulated over the years because they just want to be as "normal" as possible, I personally think it is a fair deal. I know, I know, it feels like a betrayal, but compromise can let them feelmin control of a situation thatbfeels very out of control. If a parent is super religious, or very alternative, or very hippy, the kids are embarrassed, and for some kids it can cause a real block in their relationship with their parents, which to me isn't worth it. A modicum of concession can really help here. I would also recommend finding someone to support you as well- it can feel like you are losing yourself, and you need to be clear on what you will or won't adapt. We won't ever lie, for example, or stop going to pride, or be ashamed. They don't have to feel the same way that we do though, and we understand that our love is not based on their feelings about our relationship. They will be teenagers, they are allowed to be mad at us for everything.
Finally, I'm sure this is obvious but for the love of all things holy respect the age limits of social media. It terrifies me how little people know about their kids online life and are willing to allow them to manage that on their own. Most bullying in teenage years will happen online or be exacerbated by online actions. In addition, schools have very limited powers to control what happens online- you could possibly argue that they are bringing the school into disrepute if they identify the school publicly in their posts but its tenuous. Therefore you end up in serious cases going to the police, which is an ace up your sleeve in the most dire of circumstances, but not an immediate go to I would imagine.
I personally think that you have got to approach bullying as relational. It's all about preparing children for the skills they will need to manage interpersonal relationships in the long run, and in my mind the child needs to be helped to develop the repertoire of responses, one of which might be utilising policy and adult intervention, but others may be be adjusting their own behaviour, using peer support, consolidating other friendships etc.
It isn't an easy task, and I think some adopted children can really be up against it because attachment difficulties can really start to show in friendships as children develop and go through their developmental phases. I would familiarise myself with what is normal or typical behaviour and what is not for each year group so that you have a good measure of what to expect, and always support your kid, remembering that their brains can in some cases be wired to misinterpret or misunderstand some social signals because of their trauma background. I saw someone on insta saying that their job was not to be just a parent but essentially be a replacement frontal lobe for their kids as they develop, and to help them take the signals and interpret the world sensibly, and it really stuck with me.
Hope some of these thoughts are helpful- like I say, there is what should happen, then the reality of what works and is sustainable. I'm always in favour of solutions that are sensible, practical and effective, whilst also preparing the kids for later life in an age and stage appropriate way.