Not sure which is the first instance if you're a horde player (think it's the one inside Ogrimar) but for an Alliance player the first instance that can be done is Deadmines, you need to be a minimum of level 10 and go in with a group (4 other players). The mobs reach level 20 Elite, so ideally you and your group could do with being around level 19ish, or it's going to get extremely messy for you in there :-)
You'll get the quests to enter dungeons as you naturally progress through the quest chains.
Most of the 'fun stuff' though doesn't really happen until you hit Outlands and then Northrend, but I do recommend doing the earlier dungeons as these are great practice for when (if) you reach level 80 and you're doing end game raiding.
in addition to what Boo has said, it is really worthwhile learning some professions as these can really help you through the game. depending on your character class, you need to choose the most relevant profession for example a warrior that needs mail/plate armour would benefit from blacksmithing. This in turn requires lots of ores so mining is a usual profession to go with that. Rogues wear leather so leatherworking and skinning are a good idea. Check out http://www.wow-professions.com/ as this is a great source of info on the professions and where/how to level them.
Just a cautionary note, I don't know how old you are, but if you are an adult with a family,or still at school, ask yourself if you really want to start playing this game, have a look here - www.wowdetox.com and read the stories.
good point mattie. I finally managed to 'kick the habit' last month when my credit ran out. odd thing is, I am now addicted to Answerbank :-) hehe.
seriously though, as my wife will confirm, WOW can really take over your life so take it in moderation. the key is to stay away from raiding guilds. they put too much demand on players.