*Thanks in advance to anyone who reads all this. Your advice will be much appreciated*
Little bit about me:
21, last year of university, recently started my career in insurance. My first relationship ended a couple of months ago, after just shy of 2 years. Before/during the relationship I never had/thought about game, as I was happily single and then in a relationship. Anyways, her and I got too serious, too fast, and I'm not looking to repeat that. But now I'm at a point where things are going quite well in my life, and I'm ready to go out there and meet people and have some fun, without getting serious/tied down by anything.
So last night I met up with a couple of my buddies during a "club night" pub-crawl our student society set up. I have no sort of game whatsoever, and also some social anxiety (but I've been slowly overcoming it as of late, and alcohol pretty much eliminates it). So after several drinks and shooting the shit with my buddies/their gf's/other random people who came into our booth, I had a good buzz, and was on the dance floor with some of them (as well as all the other people in the club). I was just kinda doing my own thing, then I made eye contact with this cutie, and she came over and started "dancing" (read: grinding) with me, without me doing or saying anything.
After a few songs of that, it was time to go to the next club on the list to finish off the night (it was like 12:30, the clubs here stop serving at 2). So once we were outside, where talking was actually possible, we introduced ourselves, and I handed her my phone to put her number in. Chatted on the way to the club (what majors, where we work, standard stuff) then grabbed a seat in one of the booths reserved for our club night. Talked some more about random stuff, lots of joking/teasing, physical contact, etc. Found out we have a bunch of stuff in common and clicked quite a bit, and I knew I wanted more than a one-night-stand for sure, so I wasn't going to push for her to come home with me or anything. We did a couple shots with a bunch of people in the booth, then headed to the dance floor, with her grabbing my hand and holding it while we made our way through the club.
We danced for a little while, with a couple shots mixed in, and we were having a great time. At the end of this one song, in which our physical contact was intense and hot, we pretty much simultaneously grabbed each other by the back of the head for a hot make-out session. The rest of the night consisted of us dancing and making out, and then we took separate cabs home because she lived in a different direction and had to work early in the morning. Shortly after we split up, I texted her something like "I had a great night. I hope you survive at work in the morning", and she texted back a few minutes later "I had a fun night too! I'll talk to you tomorrow

". I know (now, after my first foray today into these forums) that I shouldn't have texted her, but I didn't know that last night.
I want to share our text convo from today, which happened after I had read a bunch of stuff on the forum, and formed some sort of very basic understanding of the rules of the "game", and get some feedback/advice from some of the experienced members here. Btw, she's majoring in French, which is the reason for my French in some of the texts. I don't speak it at all, but I know a few words. Anyways, here goes:
Me (5:37 pm): Bonjour ma chere! Thanks for the French lesson last night at [club]

Where are you going tonight for round 2? Btw someone owes me a t-shirt haha! [I don't remember the details, but there was some joke last night between me/her/her friend about why I was supposed to get a club night t-shirt she was carrying around. The shirt obviously wasn't important, but it gave me something to say besides "hey what's up?"].
Her (7:23 pm): Impressive French! I don't think I'm going out ce soir though. I'm still recovering from last night!
Me (7:53 pm): Bon choix. Did I get you a wee bit too drunk? And I'm serious about the t-shirt! Otherwise I'll just have to take yours lol

. Ps, I wish I could actually speak French instead of just pretending. Wanna teach me?
Her (8:16 pm): I was a lot bit drunk lol what happened to that extra shirt [friend] had? Wasn't that supposed to be yours?
Me (8:17 pm): It was supposed to be..
Her (8:18 pm): Bien sur que je peux vous enseigner le francais! And I'll have to ask her what she did with it. [Had to Google translate this haha, it means "Of course I can teach you French!"]
Me (8:20 pm): Merci

And haha, she didn't want to give it up.
Her (8:35 pm): Not at all. She's pretty stubborn like that lol
Me (8:52 pm): I liked my shirt better anyways

[I was wearing one of my nicer dress shirts last night, and she had made a comment about how nice it felt] Since you said you love it too, let's have our first French lesson at [bar, which we both love and had talked about last night]. Toonie Tuesday! [the bar's special] I'll be there at 9ish.
Her (9:10 pm): I don't know if I can do this Tuesday

I have a take-home exam due the next day.
Me (9:19 pm): What kind of teacher misses the first day of class? :O Well if you get it done before then, shoot me a text and you can join me.
Her (9:24 pm): Our class will be rescheduled
Me (9:29 pm): Ca sonne bien mademoiselle

["Sounds good Miss"]
So, experts, what did I do right/what did I do wrong/what do I do next? Thanks in advance to anyone who responds.