I just got to the college and listened in on Torfdir's lesson and he wants me to cast and hold a ward. I guess he's supposed to cast a spell at me, but he never does and so the quest here is just not continuing on. :/

You'll have to reload and old save and do it all over again.

I've had the same issue only for me Tolfdir casts Firebolt, but just keeps telling me "You have to keep your ward up!" when the ward was up the entire time.

I suggest quicksaving right before you go into the "first class" with Tolfdir. It took me about 4 tries before it actually worked.