Gouken is a very unusual but very effective character to use.
He can use his back throw to set up an Ultra, charge fireballs, parry and set up ridiculous combos with his EX Senkugoshoha.
Where Gouken suffers is that he's easier to stun and has less energy than the rest of the cast. You also need to be fairly good at managing your EX meter, so you can always make the most from your opportunities during the match.
Gohadoken (quarter-circle towards + P)
Gouken's fireball does change speed depending on the strength used but rather, changes direction. LP travels along the ground, MP travels into the air, HP travels quite high. For the most part, you'll be alternating between LP and MP with the EX version firing a ground and air fireball one after the other.
Senkugoshoha (towards, down, down/towards + P)
Gouken's palm strike is good for breaking Focus Attack but the really good option is his EX Senkugoshoha. This is a two hit move that launches the opponent into the air, which can be followed up with the HP version or an Ultra if they're in the corner...
Tatsumaki Gorasen (quarter-circle back + K)
His hurricane kick goes straight up into the air, so it's better used as a combo finisher or after back throw rather than to move around the screen. The air version travels straight, so its best use is for escaping corners. It can be Focus Attack cancelled on the first hit but this is pointless, unless it's part of a combo.
Hyakkishu (towards, down, down/towards + K)
Gouken's demon flip move. Pressing K in the air will see him do a flip kick towards his opponent, pressing LP + LK will make Gouken throw his opponent. When you cancel into this move up close, it becomes a vital part of applying immense pressure to the opponent and forcing them to make hurried and (hopefully) bad decisions to try and escape the corner.
Kongoshin (back, down, down/back + P or K)
Gouken's counter move. Will counter high or low moves, depending on the button pressed, but is something of a risk. If you guess and your opponent doesn't do anything (or doesn't do the move you expected), you'll be left wide open for a combo of his choice.
Forbidden Shoryuken (quarter-circle towards, quarter-circle towards + P)
You shouldn't ever be using this Super, as EX moves are such a vital and integral part of Gouken's game. What you do want to use, however...
Shin Shoryuken (quarter-circle towards, quarter-circle towards + PPP)
...is Gouken's Ultra. It works as both part of a combo, after back throw and after an opponent jumps at him. These three applications alone mean it's better to hold onto the Ultra for a guaranteed moment that it will hit rather than throwing it out there and hoping for the best.
Basic Combo: Back + LP + LK > Shin Shoryuken
Advanced Combo: Jump HK > HP xx EX Senkugoshoha > HP Senkugoshoha