I got WoW, and then a month later my brother upgraded his computer so that he could play as well. We decided to start characters to play just with each other, and wanted the 'best' fit.
As you can see from the myriad of replies here, pretty much any two classes are fine for duoing PvE. The trick then comes in being good for the other aspects of the game, namely PvP and group instances.
We play on a PvE server so we didn't have to worry about getting jumped, so we went with the pair that made the most sense from an instance stand-point... Warrior and Priest.
We never really had to wait for a group for an instance ever. We already had the healer and the tank, so all we needed was any 3 classes and we were good to go. Take a rogue/warlock or hunter/hunter group and you're looking at potentially very long waits just finding the right combo of other classes to make your 5 man group work.
In terms of PvP, I'll be blunt... Prot specced warriors and holy specced priests are not the cream of the crop.
Pretty much any combo can work with the right strategy, and the most important thing is that you enjoy what you're doing. It is a game after all.
In terms of professions, if your only worry is gold then have one person be mining/skinning and the other be herbalism/enchanting. You don't need to actually be an enchanter, but disenchanting itself is a good moneymaker if you're willing to take the time to buy the junk on the AH and resell the dust.
If you want good use out of your professions, stay away from blacksmithing, leatherworking, and tailoring. Especially at early levels all 3 of those have nothing relevant to provide you that you can't just buy off the AH. Engineering provides very useful and sometimes bizarre items that you can't duplicate elsewhere. Alchemy can keep you steadily stocked in potions without needing to constantly buy them from the AH.
Maybe it's just me, but I've always found that my characters with alchemy don't mind chugging all sorts of neat stuff but my other characters just don't bother with buying them. Often they're not even for sale since they don't move very fast anyway. (Maybe that's why I chug them myself... ;))
So I would say either mining/skinning and herbalism/enchanting or mining/engineering and herbalism/alchemy.
Remember you can always choose different professions when you're high level and have the money to burn getting to the actually useful recipes at the top ends of blacksmithing and tailoring.