A Clone of Knuckles
Play through the game (with all the characters) until you get Super Sonic. In Super Sonics Adventure, go to the crashed island. There you see Knuckles and Eggman (Robotnik). After the clip, press A+B+X+Y+Start. Then choose Knuckles. Go to the same place, and there are two Knuckles! This works with other characters, too, but it's not as funny.

Clean Pause
To remove the pause menu, press X and Y together at the pause menu. This is good if you want to take screenshots.

Secret Chao Puzzle Game
After you start up the Sonic Adventure game, place a controller into the D port on your Dreamcast and have a VMS in the controller. A hidden Chao Puzzle game will appear on the VMS screen. It is a tile based puzzle game where you have to use the Dreamcast's controller to move the pieces of a moving picture around to win. After every five puzzles you solve, a little animation will play on the VMS screen.

Shortcut in VMU
When your chao says, "Chao! Which way should I go?" and the screen comes up with the choices, pick a chao and while your chao is nodding or cheering, change modes. Change modes back to Chao Adventure and your chao will be at the next crossroad. You won't get the 10 pts from the charactors this way however.

Beat Egg Hornet With One Hit
Actually, the title of this cheat is missleading. You need the Light Speed Shoes for this so you can't do it the first time you meet up with the Egg Hornet but only in trial mode. But it's still cool.
Right as Eggman is done shooting missles, rev up a Light Speed Attack. Just as Eggman lands after saying, "All system, full power!" Release the Speed Attack and if timed correctly, Sonic will hit Eggman three times!
Note: This needs to be timed very well and it may take a while to learn the timing but it does work. I have done it many times.