I fly several times each month and have used Iberia Express recently so I hope I can answer some of your questions.
Online check in - this is something that is NOT compulsory on Iberia. As long as you've filled in your passport details etc on your booking, you can just show up at the airport like you used to, go to the check in desk, hand them your passports, and get your boarding passes printed and your hold luggage checked in.
Alternatively, online check in means you complete this process on your phone or laptop - the app or website - it will assign you seats and give you a boarding pass with a barcode on it. You can print these off (after you've downloaded them as PDF's for example) or just save them on your app / phone wallet.
But like I said, Iberia Express are still 'old school' and won't charge you for requesting paper boarding passes at the desk.
After you've checked your bags and got your boarding passes, you go to security. Before you enter this area they will ask you to scan your passes.
Security is where all your bags go through an X-ray machine.
Liquids are only allowed in bottles of up to 100ml. All these bottles have to go in a clear plastic bag of 20x20cm - like a freezer ziplock bag - and onto a tray, out of your bags. They usually hand these bags out for those who don't have them already.
Any electronics such as phones, laptops, Kindle, iPads, also have to be taken out of your bags and put on a tray. Same goes for hair straighteners and similar objects.
Coats and belts need to go in the trays too. Basically, you walk through the X-ray detectors with your 'first' layer of clothing only - so take extra jumpers or hoodies off too. You stand under this detector for a few seconds as they 'scan' you for metallic objects. Flat shoes can stay on, but they will usually ask you to take off heavy boots, heels, or stuff like that.
This is where having hold luggage comes in handy - just put all of your liquids and 'sharper' objects in there if you can. I've never had an issue with nail clippers or tweezers in hand luggage, but for peace of mind I'd put them in the hold suitcase if you want.
Don't be surprised if some of your trays don't come out straight away but instead go behind a screen and a member of staff pulls them over for an extra check. Sometimes people have just innocently forgotten something in there that needs to be rescanned. Or the machine has decided to perform a spot-check on it.
After security you look for your gate and go to board there. This should be pretty straight forward, but do start walking as soon as they announce the number as some gates can be quite a trek away, or they involve getting on a train-like system for a couple of minutes, that takes you to another side of the building.
Cabin luggage allowance - Iberia Express lets you board with both a small suitcase (55x40x20 or something like that), and a handbag. Your suitcase will go in the overhead lockers, your handbag should be under the seat in front of you during take off and landing. If the flight is very busy, they might take some suitcases from the passengers and check them into the hold for free. This is usually done at the gate and they ask for volunteers first, or they might offer it at check in. Queue early if you'd rather avoid this.
Medications - I'm not an expert on this topic, but I've never had to take pills or sachets out, and I've travelled through some pretty heavy prescription cycles. It's useful to always carry a prescription just in case there's an emergency abroad, but if it's pills I'd just leave them in your bags at security. Liquid medication is different, and won't be subject to the 100ml, one clear bag rule. You can go over it for health reasons.
Due to the current situation, I'd aim to be at the airport extra early so you have time to go through all of this without rushing or getting stressed. Just make sure you've packed all your liquids properly before leaving the house - or even better, put them all in the big suitcase! - and be prepared to follow instructions, which they constantly give. Also, don't be afraid to ask. The staff are usually happy to help.
If I forgot something I'll come back and post again. Please do let us know if you have any other questions.