Help With a Guy Please?

[Nati]

I started a class in college and am attracted to this guy :]
We talked a little bit during a small class group assignment.
I contemplated approaching him in person but was worried he had a girlfriend or something.
I added and messaged him on Facebook just saying hi or whatever but it was small talk.
He's apparently single according to Facebook but seems to talk to this other girl a lot.
Should I ask him to lunch or something after class or should I wait for him to approach me?
I know I make things harder than they are but I am fairly new at this stuff and don't want to look like an idiot...
Thanks to whoever responds!