Surf is a common and effective move in singles battles. In two-on-two games, it needs to be handled carefully since it damages every target in play, whether they're friend or foe. The easy thing to do would be to pair a Surf wielder with a Pokemon that knows Protect. The more creative thing to do would be to pair the surfer with a Water Absorb Pokemon. Vaporeon and Lapras are two of the better Pokemon with the Water Absorb ability. Instead of taking damage from its partner's Surf, this ability allows Pokemon to absorb the hit and replenish hit points.
Above: Arcanine can be menacing after getting hit by its partner's Lava Plume
A similar technique can be used with the move Lava Plume. Like Surf, this move hits all the Pokemon on screen. However, it's not nearly as common as Surf, so there's a good surprise factor in using this move in two-on-two battles. The partner Pokemon in this situation should have the Flash Fire ability. A Pokemon with this ability gets a 1.5x boost to its fire moves after it's hit by a fire attack. Arcanine, Houndoom, and Rapidash are a few Pokemon with this ability. The end result is that you both of your opponents while giving your partner a nice power boost.
Two moves that are absolutely useless in singles battles but can be tremendously useful in doubles battles are Follow Me and Helping Hand. Follow Me forces the opposing team to only attack the Pokemon that executes this move. This is useful for pairing a relatively stout Pokemon like Clefable with a powerful sweeper. Clefable can absorb hits through Follow Me and restore hit points with Moonlight, while its partner can unleash offense without worry.