Which WoW class/spec has the best mobility for questing and solo'ing low level dungeons

Rogues are the quickest. With enough haste you can have a 100% uptime on 'Burst of Speed' which is a 70% movement speed. Since it doesn't break stealth anymore, you can use it with nightstalker. Not to mention the passive 15% movement speed rogues already get + sprint.

To add onto that, if you want more energy regen to fan of knives while running, go subtlety.

1) Use marked for death on an enemy (gives 5 cp)

2) Use slice and dice (grants 4 energy/sec while up for subtletly)

3) Kill the monster you marked to get mark for death back off cooldown.

4) Rinse and repeat when slice and dice is about to fall off