This is a good question for the "un-initiated". Just buy a game card and start playing. Cataclysm is coming but it's a few gigabytes of updates. Blizzard's launcher is pretty smart by pre-downloading content ahead of release, so you can just log on and play when it's officially released. So, to avoid disappointment with Cataclysm. Start downloading now.
PVP or PVE?
With PVP realms, there is a thing called "ganking", (Arqade discussion on ganking here), where groups of players just wait in ambush and just smack you senseless whilst travelling between cities or trying to complete quests for no reason apart from bulling players. If you like that sort of stuff, and want to PVP you can go this route. It does waste a lot of time when you want to go questing/farming and generally getting gear, though.
PVE has battlegrounds where you can engage in PVP play. I prefer PVE because you can get into PVP at your choosing, rather than getting ambushed every 10 minutes. PVE it's just you vs the environment. Not other players. PVE also has duelling. So, you can also pick a fight when you want to.
As a casual gamer. There is a built in quest helper, so this can speed up the grind of going around completing quests. If you're not playing constantly, then there is value in logging off in taverns and major cities, because you gain "rested XP" faster in taverns and cities, rather than logging off in the forests/roads. Rested XP basically is worth double the normal XP from kills and quests. You will level faster that way.
Best guides I have found are these:
Addons
Add-ons are enhancements to your WoW interface that can really improve game play. They have become an industry in itself, although Blizzard discourages people paying for them, you can get them for free.
You can find wow addons on Curse.
I use Auctioneer among others, but a good list of what's popular can be found here. There are class specific addons as well and you can keep them up to date with wowmatrix.
Instances
If you like raiding, I recommend that you get good at the 5 man dungeons (also known as "instances"). The key thing about instances or raids is to know the role you play in the group. It boils down to 3 different types, Damage dealers/Tanks/Healers. They're self explanatory, but Tanks are basically the one player who tries to hold the attention of the big boss you will end up fighting. This is so the damage dealers can kill it, and the healers, well... heal everyone.
Holding their attention, is called "holding threat". As the bosses or any mob you are fighting does a calculation to determine who they should kill, by estimating who presents the biggest threat. So, damage dealers tend to have high threat, because of their DPS (Damage Per Second). Healers can gain threat, because they're keeping everyone very healthy. You can improve your level of threat through talents or buffs. (A buff is like drinking a potion or a beneficial spell that gives you a special attribute for a period of time). You'll learn about which buffs are important to your style of gameplay as you progress. Finally, each instance tends to have anywhere from 3-12 bosses. You don't have to wait til you're level 80 to do raids, you can do them through level ranges of character, but the first one to do would be Stockades in Stormwind (if you're alliance), when you're about level 15, and you should go in with at least one or 2 other players of similar level. You can find dungeon groups through the dungeon finder. There is also a "raid browser". It's all built into WoW. The trade chat channel tends to have lots of offers/requests for player to join raid groups. If you do get hooked on raiding (as I expect you will), better to look for a guild to join because once you start doing 10man or 25 man raids, you need a reliable group of people to run the instances, because random or ad hoc groups can be painful, if you get the wrong group of people together.
Terminology in WoW
Basic dialog/lingo in the trade channel is like this.
LFM = Looking For More.
LFG = Looking For Group.
GS= gearscore
So, someone might say LFM Ony= Looking For More players to do the Onyxia instance. or Say LFM ICC min gs5500 = "Looking for more for Ice Crown citadel, minimum gearscore is 5500". You'll get to know the lingo as chat with people.
Trade chat is global across all cities. But only available in the cities. You can't see trade chat outside of cities.
Also, you might hear about "Gearscore". This relates to the quality of all your gear you wear. Each item you have has an "item level". The higher the item level, the better suited you are for the harder instances. The top item level (pre-cataclysm) is 264.
Gear and Items
You can acquire great gear by making it/buying it/looting it. The auction house can be good in this way to buy gear from other players. You can also buy very exotic gear from special merchants using tokens. There is a large array of currencies for different levels of gear. Blizzard are now unifying it, you got 2 basic "token" currencies, Justice Points and Honor points, (This is different from gold/silver/bronze). You earn them by doing raids or PVP. There is also some gear that you can buy from merchants, by trading other exotic items, but this is the highest level stuff (level 80).
The quality of gear you have determines whether you can do higher level instances. That's what Gearscore works out for you. There is an addon for gearscore. Get it. Players will sometime ask for people who only have a certain level of gear. They're basically trying to keep out the riff-raff, so these can be good raids to get into even though they are ad-hoc. This is common for trying to do ICC (Ice Crown Citadel instance).
At the end of the day, just do whatever you find fun in.
Best Answer
The instance you are looking for is The Deadmines.
This instance is for players between levels 15 and 21 for normal mode and level 85 for heroic mode. It is located in the southern part of the Westfall.