Here are all of the battle courses and some fun stuff to do in them.
Big Donut
Big Donut is a big circular track in the shape of a Big Donut(hence the name). Don't fall into the center of the track or you'll lose a balloon. Green shells don't do much in this track and red shells only work if your close to your enemy. Bananas, stars, and a few other items are the best.
Block Fort
This course is very fun. Red shells don't work well at all here. I always loved staying up on the blocks and firing and thousand green shells down to the low level. Then going down there and trying to stay alive. If your trying to win, green shells help you a lot here. Bananas and fake item blocks also help almost as much as a star.
Double Deck
Double Deck is always pretty fun. You'll find red & green shells next to useless here. You'll be more lucky with weapons that you deploy suchas bananas and fake item blocks. Stars and red shells help a lot at close range only. But this board is very fun because you have to drive everywhere to find your opponent most of the time.
Skyscraper
Skyscraper is a laot of fun but also very diffulcult. In the center of the course there is a deep hole which if you fall into, you lose a balloon. Also, there is an outer circular hole which you can jump over if you go full speed. Also, there are no walls so you can fall of the top of the course. Almost every item except the green shell is useful in this course.