Most Helpful Guy
You "tried"? What did you do exactly, "think" about it?
You either did or you didn't. There is no, "I tried."
How social are you? Are you a social person? Do you talk to other people? Do you socialize? Do you make new friends, or do you keep to yourself? When you're not in class, do you look for a table where you can sit at by yourself, or do you look for an empty chair in a table where people are already sitting (and ask if they mind if you sit down)? Do you start talking to people when you are next to them? Are you casual, light, and friendly? Or, does every interaction have to be meaningful, deep, and focused?
If you're naturally a social person, then talk to people (even the ones that are girls, and even the girls that you're interested in) is as easy as breathing. It's natural - it's just how you are. But if it's a major deviation or variation from your "normal" behavior, then of course you feel nervous. You never socialize with new people. This is completely awkward - FOR YOU! You don't normally do this. You don't know what to do, you don't know how people will react, you don't know what is okay and what is not. You have no experience. You don't know what to expect, and with someone you like, you want to be as conservative as possible - so you "expect" the worst! No experience or frame of reference, plus expecting the worst = nervous.
Look, you're not going to magically become social in a few days and then start talking to her.
So, here is what I would suggest:
"Test" her interest in you. Think of it like a medical exam. The doctor doesn't just start performing surgery or prescribing medication. The doctor doesn't even start with taking a biopsy. The doctor doesn't even starts off with a CT or MRI. The doctor doesn't even begin with an X-ray. The doctor starts by asking questions and looking out for symptoms. (i. e., start with the least invasive approach to test for interest... if you don't see any, STOP).
You two are already talking, which is great, because that means you don't have to break the initial communication barrier. Now, start testing for interest. Talk to her about the class material. See if she "escalates" the conversation - or if she just lets it die down until you stop talking to her (if she does this, STOP). If she escalates or continues the conversation beyond the class material, then continue to test - see if she would like to join you for lunch. If she doesn't - STOP.