I wear tight capris or knee-length shorts in the gym, or tights if I'm running outside in the winter. I like tighter stuff because a) there's no risk of anyone seeing up my shorts during a class, and b) there's no chafing. You can always wear looser shorts over top if you don't think they look good. My favorite items are by champion, under armour, and danskin.
I wear whatever t-shirts are on clearance at the sporting goods store.

I've got a lot of adidas and under armour.
I can't stand to wear cotton while working out, but if you can, it certainly makes things cheaper and gives you more options. I've seen some cotton "yoga clothes" that come in nice colors or prints and cuter styles.
I try not to shop at Target whenever possible, but they do have some really nice and inexpensive workout clothes. I've even found a few shirts at Kohl's before.