As someone who leveled a Priest here recently, stay shadow until 60, just quest or run dungeons as shadow (if people will even bring ya, Spriest isn't too good in group PvE content). Respec to
the standard 5man healing spec at 60 and start running BRD / DM East until you've got some gear, then progress to DM West and Stratholme, and finally DM North/Scholomance once you've got a good amount of healing gear. At lower gear levels, you'll also want a few stacks of potions just in case. (though really you should always be carrying mana pots no matter what)
You'll probably be able to heal the first half of BRD as shadow without problem, for the quests like Jailbreak and such, but you definitely need to be in a healing spec for anything past 7 dwarves, unless your tank is really well geared.