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Meeting girls at events: what's the next step?

Hey so I've never had a girlfriend before. In my second year of uni.

I'm all right looking. Not ugly, not too attractive either, though I think I'd look pretty good if I became slim and fit.

Anyway, so I am socially awkward and have a problem connecting with new people.

This year, I've forced myself to be more outgoing and social. Go to society events and things like that. I've met a lot of people, and quite a few attractive girls with good personalities.

Since I've never had a girlfriend before, or even tried seriously for one, I really have no idea how to go about talking to these girls and eventually going on a date. Once I get to a first date, I'm fairly certain I can manage things from there, but I have absolutely no idea how to get there.

So most of these girls I meet in events and talk to for 10-15 minutes. And then I go home and add them on Facebook the next day. I try making conversation with a question, like are you going to be coming for this future event? Then I try maintain the convo with general topics, like how's uni and stuff like that. But this always fizzles out after a while, and I simply don't know how to take the next step.

Do I simply just go for it and ask them out? But I don't think asking them out through messenger is a good idea, and I don't have their numbers. Plus, it doesn't seem like they're interested in me.

Tbh, I've actually never asked anyone out either.

Any and all advice will be appreciated. Thanks.

Also this isn't strictly just for relationships and girls alone; I have the same problem with making friends. I meet interesting people somewhere, but they already have their own friend circles, and I simply have no idea how to connect with them afterwards.