Having leveled one of each class up to or very near to level 20 I think it would be helpful to others to learn from my experiences. When I first picked a class to play I just picked and went with it, I never went back to try others until I was fully vested in the classes I had chosen originally. Skip to the end if you want some quick advice.
I'll start where I started, my druid. Looking over a description of of each of the classes on the official website I went with a druid simply because they could theoretically do everything. Plus changing forms sounded really cool. Well druids are kind of a let down when you first start one with shifting forms in mind. You are a caster basically until level 20. You do get bear form at level 10, but it isn't really all that efficient compared to sticking with the caster mentality till 20. Complexity wise druids are pretty simplistic as to the number of abilities you get over time so no fear of being overwhelmed, but there is enough to keep you busy. Lots of downtime getting mana back until 20 where cat form with talent points in the feral tree has nearly no downtime from that point on. As a caster though you nuke things, root them in place if they get too close, back off and repeat till things die.
I left my druid at level twenty-something when I got BC and started a Draenei hunter. Compared to the Night Elf starting areas the Draenei starting areas had a much nicer flow and made for easier leveling. Draenei come out to Darkshore eventually like Night Elves, but they come out 10 or more levels higher and basically skip right to Ashenvale. A huge help in leveling in what is probably the hardest set of starting area of all the races. Hunters have a little downtime for mana as they get closer to level 10, but not nearly as bad as a caster. You get a pet at level 10 and it is smooth sailing from there on out. Enough abilities to keep it interesting, but nothing overly complicated. It stays that way through to the end game. Pet tanks, you shoot, things die.
So I had leveled my druid for a while by this point and decided that my druid's cat form and stealth was really cool. So of course I decided to try out a rogue. Pretty straightforward, easy to pick up. Level up that first aid and bring some munchies because there is going to be a lt of downtime getting health back. Every one or two mobs there is a healing break depending on what you are killing. Being a rogue is all about killing them before they can kill you (or escaping if you don't like the odds). It gets a better at higher levels when you have more tricks up your sleeve. A lot of sneaking around and picking fights where you have the advantage.
By this point I had forgotten how much I disliked being a caster on my druid and since I didn't have a caster I made a mage. Similar to a rogue in the kill them before they kill you, but with a ranged advantage. There is a severe downtime waiting for mana, even worse than a rogue since all mana is regained by eating, no bandages to make it go faster. On the bright side mages make their own food and drink so no worries about running out. A lot of slow, nuke, freeze, back off and repeat till things die. One of the more exciting classes ability wise since you get so much so soon. Nothing is more fun than freezing a whole group in place and blowing them all up with an AOE.
Bore to the ing ability wise. The most complex part of being a paladin is buffing yourself, and that isn't all that complicated. Killing stuff rotates around one or two abilities on relatively long cooldowns leaving you watching your toon auto attack most the time. There is a little downtime to regain mana from time to time, more if you need to heal often. Probably a good class to start with considering the simplicity and the ability to eventually DPS, tank or heal. I am up to level 30 now and not much has changed, from what I hear from other paladins that is also the case in the end game. If you like simple though a paladin might be the class for you.
Before dual specs I was in the market to try out a healer. I had my hunter as DPS and my druid as a tank for end game content, but no healer. I am into the nature type classes so I started up a shaman and decided to level her as a healer. Very little downtime since shamans are mainly melee oriented at lower levels and even at higher levels talented in the healing tree. Lots of neat abilities to play with. I'd recommend leveling enhancement if I did it all over again. I was resto until level 75 when dual specs came out. Not perceived to be a very popular class, but it would make a great first toon.
I've only gotten my priest to level 10 since I am planning on remaking it as a Worgen with the next expansion. From priests I know and my own limited experience there is a lot of downtime for getting mana. Plenty of abilities to keep you occupied though. I wouldn't recommend a priest for a first time player.
Muhahahaha... death knights are just fun. You start at level 55 and need to have a character with that level or higher to even make one. So not a contender for a first toon even though it would make a great one. They are unique and have a bit of a learning curve no matter previous experience. Absolutely no downtime at all and after the starting area they easily overpower everything all the way to the end game. Everyone should make one at some point just for fun, even if you only complete the starting zone.
Warlocks get very complicated very fast. I've only gotten to level 17 so far and I already have more abilities than I know what to do with. Unlike other classes knowing which spells to use at what time isn't as cut and dry. There is some downtime for mana, but not as much as other casters since having a pet allows for more flexibility in saving mana while letting your pet tank. Not my first choice for a class to start out with, but it is entertaining.
Not quite as boring as a paladin, but a close second. Also has a downtime similar to rogues for healing. Of all my toons I have spent the least time on my warrior, but I am planning on starting anew with a goblin when the next expansion comes out.
Long Story Short
So if your going to make your first character or convince someone to start playing with you I would recommend an armored hybrid class that can self heal: druid, paladin, shaman. My second picks would be: hunter, rogue, mage. Leave the rest for alts unless you really think you will like it.
For races/starting areas alliance side I would recommend Draenei, Humans, Dwarves/Gnomes, then Night Elves. Horde side I've only really done the Blood Elf and Tauren starting areas and found the Blood Elf area to be very nice much like the Draenei area, but the Tauren starting area was more along the lines of the dwarves and gnomes.