This also works for installing or applying things such as spurs to boots or weapon chains or shield spikes, which you need to have a blacksmith install.
I found the newbie guides on wowwiki to be very helpful.
Thottbot is great, but I’ve been having performance issues with it (very slow) so I use Goblin Workshop as an alternate.
Joining a guild is invaluable. If you are using the default settings, you should automatically join GuildRecruitment when you enter a major city. Look out for someone recruiting and ask them if newbies are OK.
If you would consider making a character on a new server, I’m a guild master on Turalyon (Alliance side). We welcome well-mannered newbies and cater mostly to adult players, and have Ventrilo (voice chat). Email me for invite. I know all of the lower level instance dungeons well and wouldn’t mind running a Doper through (you’ll get some nice gear but more importantly some valuable group experience). Of course, feel free to ask questions here.
My best advice: use tradeskills from the beginning! Cooking and First Aid are particularly useful. Fishing’s a necessity for high level cooking and alchemy. Then, pick two professions. I’d recommend for a new player:
Herbalism/Alchemy - Great money at the auction house, useful for all classes.
Mining / Engineering - Very fun, not as much money as herbalism - ores and metal bars vary widely by server in price - but lots of interesting trinkets. Not “powergaming” by any means but very flexible. At lower levels, explosives and target dummies are very useful. This was my first tradeskill. Especially great for Hunters as they can make guns and ammo.
Or, you can go with leatherworking (esp. druids) or blacksmithing (esp. warriors and paladins) so you can make gear. It’s nice at lower levels, not so much later on.
I would strongly recommend against Enchanting for your first character as it is very expensive and doesn’t start paying back until very high levels and after a substantial (hundreds of gold) investment.
Try to get the two of you to have different tradeskills.
Don’t be afraid of the auction house. Use it, love it. If you’re melee, invest in weapons and don’t worry too much about armor. If you’re a mage/priest/warlock, get a wand as soon as possible and upgrade it - this is very handy for mana efficiency.
Many times you can get in groups with people just by asking when they’re killing in the same area.
What classes did you two select?
Thanks, everybody, for the advice! We’re learning…slowly, but we’re having fun and that’s the point of the whole thing. 
I really appreciate the offer, but I think we just want to go it alone for now. This is a social thing for us–usually in the evenings we end up with one of us (usually me) upstairs writing or playing video games and the other (usually him) downstairs watching TV, so this is a nice geeky social thing we can do together. :)Maybe when we get more comfortable with the game we’ll branch out, but for now we’re having fun exploring on our own.
He’s running a paladin and I’m a rogue. We didn’t want to mess with anything too heavily magical (though the healing has already proved nice!) and since we knew we’d be just the two of us, we wanted two classes that could fight well.
I think he’s going to take mining and engineering, and I’m going to take skinning and leatherworking.
Thanks again!
Oh–if anybody has any good advice for how to get money fast (other than doing the quests, which we’ve been doing), that would be greatly appreciated. 
You’ve already answered your question about how to make money. Mining and skinning can make you a decent amount of money, even at lower levels.
Mining and skinning are great at low levels. Skinning doesn’t do so well at high levels and then herbalism is better. You shouldn’t have to worry about that for a while though.
What you’re going to want to do after you get your leather and ore, is sell them on the auction house. There is one in Ironforge and one in Stormwind. In each one there is an auctioneer, right-click on them which brings up the auction screen. I usually list slightly less than the going rate for everything since I’m impatient and want them to sell quickly. That should get you enough gold through your 40s at least.
I’m very sure there’s one in Darnassus as well, since all the Horde capitals have an AH. Damnifino where the heck it is, though. I’ve never spent more time in Darnassus than I had to to level crafting and do the newbie quests.
SE quadrant. On the right as you go south out of the large marble warrior-full thing with the stairs.
Get the Auctioneer mod. I’m serious. Auctioneer lets you scan the AHs, takes a snapshot of what’s on the AH at that time and the price that’s being asked for it/it’s been sold for. After 2 or 3 runs you then use a feature of auctioneer called “percentless”. It takes your most recent scan and gives you a list of items that are selling for a certain percent less than their normal sell price. I’ve made mad amounts of money in low levels that way (eg. Finding a green item that normally sells for 20g going for 1g, and selling it for 15g. Someone still gets a bargain, and I still get 14g profit!). It’s also great because it gives you an idea of what you want to sell things for, for example if you have a stack of leather auctioneer will show up a box when you mouse over it that tells you what the highest, average and lowest sell prcices for a stack of that size would be, and you can adjust your sell price accordingly.
Sometimes it can tank, there’s been times when I’ve found something that’s on mad special and bought it, then haven’t been able to sell it at all. But for the most part as long as you don’t get too greedy you can go pretty well. I made 180g by the time I was level 20 with my first toon, then blew a whole bunch buying stuff that didn’t sell. Live and learn.
Rogue and a paladin… you must both be human or dwarf! That’s a good combo, he can do the melee and healing and you can DPS burst to bring the mobs down quick. Your Sap ability will help you take on groups of (humanoid) mobs as well. Pickpocket is also a fun ability to help you get extra moneys and fun things. Don’t overlook developing your Lockpicking skill, either.
Okay, money-making tips… be warned, my guild calls me the Suze Orman of WoW. 
Sell everything. Nothing is vendor trash if you’re broke. Even a few coppers will add up.
Something that you’ll be looting off mobs at your level is Light Feathers. They drop off most birds occasionally. Save them and sell them at the Auction House (AH). Mages and Priests use them in spells and they usually sell well. Also, these three “trash” items: Soft Bushy Tails, Torn Bear Pelt, and Small Furry Paw - occasionally drop off mobs. They are used for Darkmoon Faire turn-ins. If you have space in your bank to save them until 12/4 when Darkmoon Faire comes back to Elwynn Forest, the price for them will go up on the AH.
http://gathereraddon.com/ – this addon helped me a lot when I was starting out as a miner. It tracks herb, mineral, and treasure sources in the game. The “treasure” sources include all the chests you find and lots of other items which you haven’t encountered yet in the game. The latest edition also tracks Fishing areas: schools of fish, wreckage, and elemental water. You can also periodically upload your data to their site and download a new file which includes the sites where other Gatherer users have found these items!
Don’t forget to auction off the stone you aren’t using as an engineer. You may want to farm especially for the coarse stone which comes out of tin veins when you get to that level. It’s fairly scarce, and used in lots of crafting items, so it usually is selling high.
Fishing is a “free” gathering skill, and there is a lot of money to be made from it. In lower levels, concentrate on fish needed for alchemy, such as oily blackmouth and firefin snapper, which sell better than food fish. (Fishing is boring as hell, but it’s free money.) If you want to go to The Barrens, deviate fish is a big big seller, and a low level fish to farm. Fish in wreckage on the coasts to get stuff like lockboxes and other neat things.
Cloth always sells. It may not sell at a staggeringly high price, but all it costs is the time to farm it off the mobs. Collect cloth and sell – linen at lower levels, then wool, silk, mageweave, as you progress. (Remember, only humanoids drop cloth.)
Create a “bank alt” – level a character to 5 to learn (dis)enchanting. (I recommend a human because they’re easiest to run to a major city at a low level.) Now, with a disenchanting bank alt, you can disenchant any green or blue Bind on Equip item that you cannot use and which won’t sell on the AH. *Enchanting mats require no deposit at the AH, so you’re essentially auctioning them for all profit. * You can continue putting them up for auction over and over if they don’t sell, and it won’t cost you additional AH fees.
Those tips should help you get started as lower level characters. When you get a bit of gold to your name, you might want to check out the Auctioneer addon (www.auctioneeraddon.com) and start “working the market.”
Let me second the recommendation for Auctioneer. Even if you don’t use it for speculating at the auction house, it’s still great for telling you what the current going rate is for things you’re trying to sell.
I also recommend taking two gathering professions until you hit level 40 – either skinning/mining or skinning/herbalism. On most servers it’s very hard to turn a profit with any of the crafting professions until you start acquiring rare recipes in the 200+ skill range. There are just too many other low-level players making the common stuff and flooding the market with it.
Besides, if you’ve got a nice wad of money sitting in the bank it doesn’t take long to retrain in a crafting profession. I’ve retrained to 200+ in both engineering and leatherworking in just a couple of hours spending only about 20-30 gold in the process.
Thanks! I’ll check out Auctioneer when I get a little more proficient with the game.
BTW…one more newbie question: Does Teamspeak cost money to use? We downloaded the software (Mac version) and got headsets, but the thing nobody seems to explain very well is how to connect and whether it costs money to connect. Are there any free servers? Thanks again for all the help!
Teamspeak is free, but AFAIK you must host your own service. I have to say Ventrillo has been better (as a user) in almost every way.
We just found a way around it–the spouse pointed out that iChat has audio chat capability, so we’re using that and it’s working out great for us. So I guess I don’t have to navigate Teamspeak after all. Thanks, though!
The Newbie forums. Read both the US and World of Warcraft ones. Some good stickies in both.