Just do quests your level. Green and yellow ones. There are very few quests with items worth your while, unless you intend to level up very slowly. Otherwise you'll outlevel the rewards fairly quickly and should focus on gaining levels.
If you choose to do so, check the AH once in a while to try to keep your equipment up to date and check
WoW Loot to know what you need from which instance. Again, don't run them thousands of times for items you'll outlevel quickly. Just get the basic and move on. You can manage most quests with average greens and even underpar items.