Mario:

Regular Attacks:

Punch: Press A right before the first punch.
Hammer: Press A in the middle of the first swing.
Shell: Press A just as Mario kicks the shell at the enemy.

Magic:

Jump: Press Y just as Mario lands on the enemy
Fire: Press Y repeatedly as many times as you can
Super Jump: Press Y just as Mario Lands on the enemy

Mallow:

Regular Attacks:

Punch: Press A right before the first punch.
Staff: Press A as the stick hits the enemy.
Cymbal: Press A right as the 2 cymbals meet.

Magic:

Thunderbolt: Press Y as soon as the lightning hits.
HP Rain: Press Y just as the shower ends.
Psychopath: Press Y as the 2 spotlights both hit the enemy.
Shocker: Press y right as the bolt hits.
Snowy: Rotate the Control Pad as long as you can.
Star Rain: Press Y as the star bounces.

Geno:

Regular Attacks:

Punch: Press A right as his arm hits the enemy.
Gun/Cannon: Press A right before it shoots.

Magic:

Geno Beam: Hold Y until 3 stars appear.
Geno Boost: Press Y right after the arrows dissapear.
Geno Whirl: Press Y just as the discs hit the enemy.
Geno Blast: Hold Y until 3 stars appear.
Geno Flash: Hold Y until 3 stars appear.

Bowser:

Regular Attacks:

Claws: Press A right after Bowser slashes the enemy.
Chomp/Link: Press A right as the link/chomp hits the enemy.
Glove: Press Y just as Mario hits the enemy.

Magic:

Terrorize: Rotate the Control Pad as long as you can.
Poison Gas: Rotate the Control Pad as long as you can.
Crusher: Press Y just as the rock comes out of the ground.
Bowser Crush: Press Y repeatedly as many times as you can.

Princess Toadstool:

Regular Attacks:

Glove: Press A just as she slaps the enemy.
Fan/Parasol: Press A just as the fan/parasol hits the enemy.
Frying Pan: Press A right after the pan makes contact with the enemy.

Magic:

Therapy: Press Y just as the stars disappear.
Group Hug: Press Y just as the stars disappear.
Come Back: Press Y just as the star hits your ally.
Sleepy Time: Rotate the Control Pad as long as you can.
Mute: Rotate the Control Pad as long as you can.
Psych Bomb: Press Y repeatedly as many times as you can.