A duo levels faster than two solo players, as long as the pair covers each other's gaps. Here are the combinations we'd pick.
Tank and healer pairs
Warrior + 
Priest: the classic. The Warrior holds everything, and the Priest shields and heals. Downtime almost disappears, and you can take on elite quests early.
Paladin + 
Shaman: two sturdy hybrids that heal each other. Dwarves can now be both, so this pair works on the Alliance side for the first time. On the Horde side it's an Undead Paladin with any Shaman.
Two self-sufficient classes
Hunter + 
Druid: the pet tanks, and the Druid switches between cat damage, bear tanking and healing.
Warlock + 
Mage: lots of damage, and the Mage makes the water. Watch for pulls that reach you.
Two of the same
Two Hunters or two Warlocks with two pets is very safe. Two 
Rogues are great on PvP realms but eat a lot.
A tip for any duo
Keep your levels in step. Shared quest progress and kill credit only help if you're on the same quests. One of you leveling ahead on the weekend undoes a lot of it.