This answer will refer specifically to Mists of Pandaria end game (level 90 content). As of writing, World of Warcraft has been updated to 6.0 and a new expansion has been released - Warlords of Draenor, which provides a slew of opportunities to earn more powerful gear at level 100.
Probably the fastest way to earn Mists of Pandaria end-game gear is to make a trip to the Timeless Isle. Timeless Isle is an island with a lot of treasures on it. These treasures, when looted, will give you some currency that can be used on the island as well as a chance for item level 496 loot tokens. Making a quick circuit around Timeless Isle is a fast and efficient way to obtain entry level gear.
There are several guides for doing this. Here is one of the more popular ones: Timeless Isle Hidden Chest Guide!
Basically, the idea is to loot all the chests. The map shows the location, and divides them into groups (higher number chests have a better chance to contain tokens). I really recommend checking the linked content, as it has a lot of information that is difficult to summarize.
Hitting all of the chests is likely to earn you numerous item level 496 pieces, as well as enough coins to buy another token or two and a guaranteed item level 535 upgrade token from the blazing chest (marked 7 on the map). This should greatly boost your item level. Also note that the Amphitheater of Anguish scenario rewards an item level 450 weapon.