Successfully complete the game to unlock the New Game Plus mode.
Successfully complete the indicated task to unlock the corresponding character:
Cream: Get Knuckles on your team, then go to Cream at the center of Green Hill Zone. Find Cheese in the tower to the southeast, and Cream will join your team.
E-123 Omega: Successfully complete the side quest in the northeast of Metropoplis. Ascend to the top, and have Tails fly to the west. E-123 Omega will be in an underground room.
Defeat the first Kron you encounter. Save the game, then go south to an opening. You must fight another Kron. After you defeat him, with Amy in your team, go to the opening west from that area. Once you enter, bash the boxes with Amy's hammer. Go forward, and ignore the Kron. If you get attacked by him, just flee. Go to the next room. Continue until you reach the ruler, then talk to him.