As for what moves each would have
I think there should be ten distinct TMs, each easily identifiable by color.
Red: Flame Thrower - like a longer ranged fire flower.
Orange: Focus Blast - a chargeable energy blast that can do massive amounts of damage, if charged enough.
Yellow: Thunder Bolt - behaves sort of like Pikachu's thunder bolt, but it can paralyze opponents temporarily.
Green: Bullet Seed - behaves sort of like a scope gun but with more ammo and no charging.
Blue: Bubble Beam - shoots a flurry of bubbles and foam, that can trap foes inside giant bubbles.
Cyan: Ice Beam - shoots a beam, like R.O.B's laser, that will freeze opponents. It has cooldown in between shots.
Purple: Toxic - shoots a short range needle that badly poisons opponent.
Pink: Teleport - behaves like Zelda's teleportation move.
Brown: Earthquake - tremors the ground underneath, burying anybody in proximity to the player.
Gray: Swords Dance - a flurry of slashes is released, behaves like a more powerful and longer version of Meta Knight's jab move.
I think this lineup is good because it encompasses a variety of types. I tried to make them all able to work for any character by making most of them projectiles or visual effects. Please share your own ideas!