To choose your alternate paintjob in Main Game, Practice or Boss Attack mode, press the R button at the plane select screen.
To select the alternate color in Death Match, hold the R button while selecting one of the four original planes.
Bonus Plane
At the "PRESS START" screen, press C-Left, C-Down, C-Right, C-Up, C-Left, C-Right, C-Down.
Extra Planes in Battle Mode
To earn six new jets, beat the first two stages to earn the MIG 25 and MIG 31, complete the third and fourth stages to earn the F-22 and KFR C2, and finish the game to earn the EFA and RAFALE. Good luck!