The class that makes the game easy mode is DK. You get to skip 55 levels for one thing. It is not hard to be half decent at blood. You can solo most appropriate-level 5-man content if you are geared well. You are stupidly hard to kill in PvP even with nerf after nerf after nerf.
If you like ret, paladin is a close second. Once upon a time, protadin and holydin were awesome leveling specs. Now they are just meh. Prot is just really really sad from 85 on up. Nothing can kill you but it takes all day for you to kill anything else. And holydin has no AoE.
My two most-played characters are a priest and rogue. I wouldn't describe them as fast to level. They're not bad though. A rogue can stealth past mobs which is often useful when questing. A priest in disc spec can handle a ton of incoming damage and solo things you wouldn't think a healer could solo.
Lets look at classes:
Mage - 3 different specs, all viable in raiding on a average level so you can switch between them when you get bored with rotation and spells.Mages are always in demand for pugs/raids so this is good choice.
Priest - you have only one spec to play with, so it short your choice to shadow if you dont like healing. Some people who played spriest will tell you it is boring class but this is just their opinion.
I suppose any dps class is kinda easy to "hide" in a raid because you hardly can cause a wipe...which as a tank or healer you can. However if you even have the remotest inclination to try healing...level that priest. Also atm in my Battlegroup, healers are totally in demand in LFR and LFG, guranteed short queues.