Dressing Your Avatar
Here is a great link that details all the erm, details of dressing up.
You can buy items with gold in the shops, marketplace, or some quests. Or, you can purchase items through monthly collectibles or Gaia Cash.
Clicking on your avatar in the upper-left corner of the screen will direct you to your dress-up where you can change your avatar.
You can reach your inventory, hover over the My Gaia button and then click 'Inventory' from the drop down. You can also get here by clicking My Gaia, and then selecting Inventory from the options along the top of the landing screen.
Changing your Base
Once your base is created, the only way to change it permanently is to purchase these changes at the Salon or at Skin Tyte.
The Salon will offer you hundreds of different hairstyles- the vast majority of which were not available to you when you created your avatar. You can also change your default eyes here.
Skin Tyte sells potions that will allow you to change your skin tone to any of the starter tones, in addition to some special potions- such as dark elf and centaur bases.
Notes regarding avatar event skins (NOT item skins, such as Superior Form)...
During a number of seasonal events since 2005, special skins were available, including
Grombie/Zombie Skins (2005 Halloween Event)
Zurg (Alien) Skin (2006 Christmas Event)
Vampire Skin (2007 Halloween Event)
Some of these items are available as items now, while others are now available as potions that will change your base. You can remove any of these effects by removing the item or drinking a human base potion.
Dream Avatars
A dream avatar shows off your sense of style even before you can afford to spell style. Another advantage is allowing other Gaians to view your wishlist quickly by simply comparing your dream avi to your current, real avatar.
There are two ways to play with items you do not yet own. IF the Avatar Builder is working (a big IF, and if it worked yesterday it may not work today) that is the easiest way to browse Gaia's huge database of items by color, position on avatar, and price. It also gives you a view from every angle that you can copy all or part of, or click "copy" to get a code for the images and list of items. The Avatar Builder also tells you how much each item costs on the Marketplace (beware, that cost may be misleading) and a compiled cost of the whole avatar.
However, if that feature is not working today (or this month or whatever), you can still play by going to your avatar dress-up page and putting in the name of the item you want to play with in the search bar and clicking "All" button above the inventory. If you put in too broad a term (like "red") you will only get 600 items and a warning to narrow your search. If you're not sure what to search with, start by searching the marketplace where you may have better luck. The advantage of marking items you like to "add to my wishlist" is that makes them easier to find again. Unfortunately you can only equip one pose per item this way.
*Special thanks to Hachi-Kagomi and Ned-san and Daypaw.