Get Extra Hard Mode

Open the V-Hard mode by first picking your game and then in the Game Settings menu press and hold Z then press L, L, A, B 

OR you can finish the game using the Hard difficulty setting.

Super Hard Mode

At the game settings screen hold z and press L,R,A,B.

Get Mewtwo's Stage

Complete the game using the S-Hard difficulty OR go to the trainer select screen, HOLD Z then press B, Up, L, B, A, Start, A, Up, R.  Now player 2 will be mewtwo and the 1st player Ash

Get Marathon Mode

Go to the title screen and Hold Z then press B, A, L, L.  Now the speed can be set up to 99.  You can also use time zone and get 10,000 points to get this.

Get ?? Trainers

Unlock the ? trainers by holding L + Z + R on controller's 1 and 2 at the trainer screen.  You can also finish the spa service game to get this

Get Badges

Get the badges in bold by defeating the characters to the left

Brock Boulder 
Misty Cascade 
Lt. Surge Thunder 
Koga Sould 
Erika Rainbow 
Sabrina Marsh 
Blaine Volcano 
Giovanni Earth 

Unlock Options in Puzzle University

Go to the title screen and press and Hold Z then press A, B, R, A, A, B, R, A

How To Make Jigglypuff sing

Go To Marathon or Time Zone, and press L or R until the music CHANGES then press L or R once more and quickly get a combo or a chain.
This will get you somewhat more points.