Give us some background. Are you Horde or Alliance? What is your class and spec? What is your previous MMO experience? Are you finding the quests easy or difficult?
Why does it matter?
The first couple of dungeons play slightly differently. I always found the first Horde instance slightly more forgiving and even if it wasn't, it was right there in Org.
Some classes and specs may be slightly less gear dependent than others.
I never played the FF series MMOs, but I gather that WOW (especially today) is probably easier on the new player. If you've never played an MMO before, you might find your first dungeon run a little confusing. Many players know the dungeons a bit too well and have so-called heirloom items that give them comparatively good gear for their level; they tend to rush through things.
If you are finding questing easy, welcome to the
jungle dungeon. If you keep pulling three mobs and getting crushed, you should probably read up on threat and aggro. Not understanding those things is a quick ticket to getting the boot from your group if you cause wipes.
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Once you are level 15 press the letter I on your keyboard, that will bring up the queue for random dungeon with everyone else of the right level currently looking for a group.
For the most part, this is really all you need to get started. The other things may just help us give better advice about how to do things more smoothly. You can help things by reading your dungeon journal before you enter the first dungeon.