Well I had a date last night and I thought it went pretty well. When I walked her to her door I told her to call me today and she didn't. So, should I just give up?

No, don't give up, she might have had to do something, and never got around to calling. Wait a week or so, and if she doesn't call then, you can either call her, or give up.

You probably shouldn't have told (did you tell her or ask) her to call you. She had probably decided whether or not she was going to call you back long before you got to the door. By demanding/requesting that she call you, it might seem like you don't think she would call on her own. She could see this as insecurity (which I think it is), and think a little less of you. Hopefully she isn't some superficial bitch, though.

I think my exact words were "Hey, call me tomorrow or something." She's not superficial though, I do know that much. Well I have class with her tomorrow and I wonder how I should approach her, any ideas?

Just talk to her normally. Maybe ask her if she had a good time, unless you established this already. Just because you went out with her doesn't mean you need to act differently around her, in fact, it probably means the opposite of that. She went out with you because she likes you, so changing your attitude toward her or something like that would be idiotic. Hope this helps.

Don't push her into a corner about asking her about it. She coould have simply forgotten. Be nice and be understanding.