Crying in confession is completely normal. As is laughing and lots of other emotional responses. I work in a church. We get shouting (which I have to make special efforts to try to not hear, e.g. putting my fingers in my ears and humming!) - both angry shouting and hard-of-hearing shouting; crying; singing; laughing - all sorts.
Personally, I don't cry very often, but there's one priest who nearly always makes me cry and he's one of my best friends. He just knows what to say to make me realise what I've actually done and be really sorry for it. I don't feel embarrassed about it, I know he's seen it all before. And it can be a sign of true repentance, which is good (it can also be just a sign that someone is a bit over-emotional, tired, hormonal, whatever - so I'm not necessarily equating the exterior sign of crying with repentance, but it can be).
If you are stressed and nervous about it, I would suggest preparing what you want to say beforehand and sort of rehearsing it in your head. You can use your fingers to help 'count off' the things you had to say so you don't worry about forgetting something, or it's fine to write them down as well - as long as you're careful to destroy the paper afterwards. It might help to go through one of the online guides which can walk you through the process, or say to the priest as you begin that you're quite a new Catholic and you'd like some help to be guided through the process. If the confessional where you are doesn't have an act of contrition printed on the wall, you can bring your own in on paper unless you prefer to make it up.
It might also help to take some time to decide where/who you want to confess to - assuming you're in a position to be able to travel a little and have the luxury of choice. You have the basic options of your local parish where the priest knows you and is able to guide you personally, or a more anonymous setting like a cathedral or a prayer group where several people are hearing confessions at once. For a regular thing, going to the same priest can be really beneficial and help us to grow, but if there is one particular thing that you're really embarrassed about, it sometimes help to know that - even on top of the seal of the confessional - the priest is a complete stranger whom you'll never meet again.
The more you go, the easier it gets! Why not try going once a month or so and get into the habit, it becomes more and more natural over time.