Play as Akuma:

At character selection screen highlight either Magneto, Juggernaut, Dhalsim or M.bison and press Up and Akuma will appear.

Play as Apocalypse:

Beat the game on a difficulty setting of 5 or higher without continuing. Go to Battle/V.S. mode and highlight Akuma. Hold Select for 5 seconds, then press any button without releasing Select.

Apocalypse's moves:

Play as Chun-Li (Street Fighter 2):

Highlight Chun-Li at the character selection screen press and hold SELECT for 5 seconds and press any button to play as Chun-Li with the outfit from Street Fighter 2.

Extra Options:

At the Main Menu (where you choose Arcade Mode, Vs, etc.), quickly press: Triangle, Triangle, Left, Circle, L1. If done correctly an option screen will appear. From here you can change to Original mode, which enables tagging in VS. Mode only. You can also set your Power Meter to always be charged.
If the above code doesn't work for you, try Triangle, Triangle, Right, and L1 (works on some versions of the game).

Tag Team in VS. Mode:

After switching to Original Mode, start a two player Vs. game. Choose your characters and have your opponent choose the same in reverse. For example, if you choose Ryu and then Ken, your opponent must choose Ken and then Ryu. To switch characters during play, hit L2 and R2 simultaneously.