Level with a pet if you can because it will help keep you safe. If your pet is tanking all of the damage from the enemies you’re fighting, you won’t have to. The other reason is that pets are easier and cheaper to heal than you are. If you don’t have a pet, make sure you know how to use your defensive abilities. You can’t level quickly if you are dying all the time.
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Usually it's best to avoid a lot of instance runs. It is too difficult to account for bad groups, and some instances are too far away with too few quests. Since the experience is very good, it's a good idea to clear nearby instances which have a lot of quests once (Gnomeregan, Deadmines for Alliance). Distant instances with few quests should be avoided (Deadmines for Horde). Low level instances can be cleared by paying or befriending a single high level character. However, since patch 3.3 a new Dungeon group finding system has been implemented, as well as a direct port to dungeons (when entering the group you get ported to the dungeon and back to the same location if you leave it) and thus the travel time to dungeons has been greatly reduced. This makes it a good idea to complete quests or grind while you wait for a group to become available.