I've never heard anything to suggest you have to decline invitations to have values. Anyway...
Have you ever said anything to her to indicate that you like, or are interested in, basketball? If so then she is going out of her way to contribute to your lifestyle which is good. Either way she is trying to spend time with you so as long as you don't completely hate the sport or honestly do have other plans that day I say go. Unless, of course, you're not really interested in this girl in which case it would be a bit shitty of you to lead her on and take gifts from her when you have no intention of giving anything in return.
As for starting to like one of your girlfriends, great man
Just be sure you know what you really want. Be true to yourself. If what you honestly want is an exclusive relationship with this girl (which is what I'm getting from what you said so correct me if I'm wrong) then there's nothing wrong with starting to like her.