Complete the tutorial, then enter one of the following codes at the main menu to activate the corresponding cheat function.

Iron Spider-Man suit
Press Up, Right(3), Left(3), Down, Up at the main menu.

Negative Zone suit
Press Left, Right(2), Down, Right, Down, Up, Left at the main menu.

Scarlet Spider-Man suit
Press Right, Up, Left, Right, Up, Left, Right, Up, Left, Right at the main menu.