To use cheats such as alternate modes, And Knuckles and Debug, you must go into Mania Mode and hover over the 'No Save' icon on the right. Next, press Triangle (or Y on Xbox, or X on Switch) to activate these options. Here is how you use each mode.
And Knuckles will add Knuckles as a second character, even to Knuckles!
Debug is activated by pressing Triangle (or Y on Xbox, or X on Switch), then the jump buttons to place and rotate through options.
Mania's move is a Drop Dash (hold jump in the air), Sonic CD's by holding Up then jump, and Sonic 3 and K Insta-Shield by pressing jump in the air.
Finally, there's Level Select, which interestingly works in a variety of ways and differs from platform-to-platform.
Sonic Mania PS4 Level Select: Select the No Save option on the far right of Mania Mode with Debug Mode enabled, and press Options, Triangle and Square.
Sonic Mania Switch Level Select: On the title screen (back out of the main menu to reach it) press Y + B buttons together.
Sonic Mania Xbox One Level Select: Select the No Save option on the far right of Mania Mode with Debug Mode enabled, and press the Menu button, Y and X together.
Sonic Mania PC Level Select: Select the No Save option on the far right of Mania Mode with Debug Mode enabled, then the Q and A keys together.