Here's all of Godzilla's moves:
Short Punch - II (release immediately)
Power Punch - II (hold longer)
Duck Punch - Down + II (release immediately)
Duck Tail Sweep - Down + II (hold longer)
Dash - Down + I (continues as long as Down is held)
Head Butt - II during a dash
Jump Punch - II during a jump
Jump Kick - Down + II during a jump
Bite - Down + II in a clinch
Shoulder Throw - Up, Towards (or Away) + II in a clinch
Body Glow - Down, Down-Towards, Towards + II in a clinch
Tail Attack - Down, Down-Away, Away + II in a clinch (invincible attack)
Fire Radiation - Down, Down-Towards, Towards + II (continues as long as II is held, press Up/Down to change direction of fire radiation)
Hyper Radiation Rays - Up, Away, Towards + II
Jump Radiation Rays (1) - Up, Away, Towards + II during a jump
Jump Radiation Rays (2) - Up, Away, Towards + I-II during a jump
Roar - Up, Away, Down, Towards, Up (seemingly useless and I can't pull it off with any sort of consistency)
I take it you don't have a manual. I'm too busy lazy to type up the other character's moves just this moment, but if I get a chance, I'll take it to work next week and scan in the manual.