In order to play the later tracks in two-player mode you first have to reach them in a single-player game. Then lose a race and enter your name as "ATR". Now go back to a two-player battle and you can select any of the hidden tracks.
Go to battle mode and select Forest World. Choose track two and race in a formula car. Let player one win all the races, and when you get to the title screen, select battle mode again. You should now be able to select the Space World.