(Please note that the recommendations below are largely made for in-game play. A Goodra in a competitive environment will likely make use of different moves to accommodate for smarter, more adaptive opponents. If you have any move sets of your own, toss 'em in the comments.)
Goodra's a bit of an evolutionary oddball. It starts as a Goomy, evolving into a Sliggoo at level 40. Okay, so far, so good. Once you have your Sliggoo, keep training until you get it to at least level 49. Once that's done, you need to find a Route that is currently rainy, then level it again. At a minimum of 50, Sliggoo will turn into a Goodra. Note that the Route MUST be rainy! In-battle effects such as Rain Dance will not evolve a Sliggoo into a Goodra!
Electric, Fire, Grass, Water.
Dragon, Fairy, Ice.
For the most part, Goodra is a behemoth. Its HP is pretty good, its Attack is enormous, its Special Attack is even better, and its Special Defense... woof. Woe unto any pokémon that tries to chip through that Special Defense. Even its Speed is a little better than you were probably expecting. Goodra's only downfall is its base Defense, leaving it rather vulnerable to any physical moves, not just ones that are super effective.
Sap Sipper: Grass-type moves used on Goodra will restore its HP rather than doing any damage. I highly recommend picking up a Goodra with this ability, as it effectively negates grass-type moves against your pokémon.
Hydration: Cures any status ailments incurred on the pokémon while it's raining. This can be handy if you intend to use a Goodra with Rain Dance. (I still prefer Sap Sipper.)
Gooey: When hit with a physical move, enemy pokémon lose a bit of speed. Helpful, but your Goodra is strong enough that it can bring down most enemy pokémon in one hit anyway. Gooey is a hidden ability.
Goodra is an absolute powerhouse, and you should probably use it as such. Slather it with hard-hitting moves, STAB and otherwise, which take advantage of either its Attack or Special Attack, since both will turn out quite good. Outrage is a spectacular STAB move in Goodra's hands, and beyond that... well, you have lots of options. Power Whip is a good naturally-learned move, Return is good with some work to up its happiness, Thunderbolt can be surprising, Sludge Bomb, Flamethrower, Earthquake... frankly, you should fill up Goodra's slots with moves that aren't covered by the rest of your team. So long as you don't double up with moves your Goodra will excel.
The other option, open only to Hydration Goodras, is to make use of Rain Dance. If you think an opponent is going to be problematic, you can set up Rain Dance, spend a turn or two attacking, and then use Rest to fully restore Goodra's HP. Rain Dance will dispel Sleep status at the end of the turn, reliably bringing your Goodra back online for the next turn. The other two slots can go to whatever moves you like, though I still recommend that Outrage be one of them.
EV / Super Training
You can train Goodra's offensive capabilities if you want, but it's probably wiser to focus on its deficiencies - in this case, either Defense or Speed. I recommend Defense, myself: if you can manage to boost Goodra's Defense to the same levels as its Special Defense you'll have a pokémon that the enemy will have a very, very difficult time taking down, even with super effective moves.
Catching a Goodra
As mentioned above, Goodra is a little tricky to obtain. The first step is to capture either a Goomy (found uncommonly on Route 14, just south of Laverre Town) or a Sliggoo (found uncommonly on Route 19, south of Couriway Town). Then you must level your pokémon up to a minimum of level 49 and obtain its next level on a rainy Route. Remember, moves like Rain Dance WILL NOT evolve your Sliggoo into a Goodra! You have to move from Route to Route until you find one that's currently rainy!