Hi fellow members of r/seduction!

I've [26M] recently moved back to my parents after finishing my college degree [Engineer] while I wait for a job. I'm actively looking for it but in the meantime I wanted to have a decent life in here. I'm living now in a 20k people town, the closest univeristy is at 50 km (I'm European). Well, I've decided that I'll hit the gym, enjoy my hobbies and improve my english while I'm here so my cv gets better.

Well, this afternoon I've started my Bodypump classes on the gym, I really love that, I used to do it 8 years ago ( the coach remembers me) and I was really into doing it again. When I was signing the papers for the membership (a lil messy hair and a regular t shirt nothing fancy) a really cute girl [7/10, awesome face average rest] entered the gym and I thought that it would be great if she was going to take my class too. Well, she did, when I saw her it was like *Nice* but I was busy talking with a guy I know that works there too so I left it pass.

Here's the situation, I'd like to talk to her and try to get to know her outside the gym environent but it doesn't seem easy because today for example she left quite fast after the class. I've thought that one of these days I could just try to go back home with her and try to get the most out of that moment but at the same time I don't wanna lose the *NEW* effect that I might have. I've considered talking to her before the class starts but I don't wanna get other classmates listening or similar since those are going to be with me for a lot of time and don't wanna have them conditioning any of us. I think we've crossed sight/eyes a couple of time but nothing remarkable (I can notice the obvious).

Well, for more context I've been letting my hair grow because I wanna get it to a ponytail in the close future, but now is a lil afro and I don't feel entirely comfortable with it, on the other hand I have really great blue eyes and a decent body tho she can lift x3 what I can (not much now btw), but since I'm new that's okay. I know I'm a good talker and decent picking up girls but I've lost my style since I left a LTR 2 months ago so I'm relearning, I know how to escalate once I've got the date, but I've never been too awesome on the previous steps tho I could handle myself, now most of the "fear" is the akwardness that could come, but thinking positive I could only be hitting that gym for 3 months or so so time I waste is time I lose if things go well

So... the question, should I try to approach her and try to initiate a talk? If I should (I think I do) where would you do it? Aaaand girls of reddit, what do you think of this, are you open to be initiated on the gym?

Thank you!!

More info can be provided and added on demand on future edits if needed