Excellent advice is given above. What follows are my own opinions, which are worth whatever value you choose to assign to them…
You can easily solo or duo from level 1 to 85, without ever even entering a dungeon or a battleground (team pvp play).
I strongly suggest you pick a “normal” or a “roleplay” server to start on, and Cairne would be a great choice if you choose to go Horde, because of the very helpful and friendly doper guild, Burning Dog Legion on that server. I strongly advise against starting on a PVP server. Once you learn the game a bit, that may be something you’d like to try, but it’s a different, and considerably harder, way to play the game.
Don’t bother spending any gold on buying equipment/armor while leveling up. You’ll be fine with whatever you get from loot/quest rewards, and you’ll want that coin for other things later…like mounts/riding skill.
Do spend gold on outfitting your toons with the biggest bags you can afford. More bag space means fewer trips to town to sell your loot, and more time spent playing.
The first thing you should do after entering a new zone is got to the closest fligtmaster, and click on him to get credit for the flight path. Once you’ve done that, you can fly on the giffon taxis to go there again.
If you’re going to be duo with your wife, you probably want to go with the same races. Otherwise you’ll be in different starting areas, and it might be a little challenging to get together at low levels.
For classes, I’d go with a Tank/Healer duo. Together, you can likely handle any “group quest” in the game outside of dungeons/raids without need for anyone else. A pair of paladins or druids gives you both maximum flexibility, as those classes can fill all three roles–Tank/Heals/DPS, but the downside is you’ll both want the same gear when it drops.
Really, any combination of classes is viable as a duo. Play whatever you like. Maybe try different races/classes/specs on multiple toons. You get up to 10 characters per server (realm), and 50 maximum across all realms.
You might want to look into the “refer a friend” program. I do not know exactly how it works, but it provides certain advantages when teaming with your “friend”. It’s really intended for an established player and a newbie, but might be worth considering for two newbies.
Once you get started, any question you may have about the game can probably be answered by someone right here on the Dope, in the massive WoW thread.
Never, ever, under any circumstances should you give your passwords to anyone other than your spouse. Blizzard employees will never ask for your password.
Never, ever, under any circumstances should you buy in game items or services (ie, leveling services) from out of game sources. Not only is that a violation of the EULA, subject to banning, but you’ll also get your account stolen.