First a quick look at the skills you get as a starting priest
1 - Smite: Basic holy damage ranged attack. 2.5 Sec cast its slow.
3 - Flash Heal: 1.5 Sec cast and hits decently hard at your level. useful in case you actually need healing.
4 -Shadow Word: Pain: An instant damage over time that helps contribute to kills. Shortened to SW:P.
5 - Power Word: Shield: Your bread and butter of of invincibility at lower levels. Prevents you from taking a good amount of damage. It has a debuff that prevents it from being used again for a few seconds after its cast.
7 - Inner Fire: A self buff that increases your spellpower, the effects of all spells cast, and gives you more armor.
8 - Renew: An instant heal over time, HoT, that heals about as much as SW:P causes damage.
9 - Mind Blast: A heaview hitting spell than Smite it also has a timer on its use, also comes with a longer range.
Someone spent time working over the lower levels of this class (unlike some others). The Smite spam is boring but doesnt last too long, just enough for you to figure things out. Flash heal is redundant with the instant health recovery but every so often you end up in some weird fight that you cant get out of.
So for casting
1: Smite! :)
4: Smite, SW:P, Smite till toast.
5: Powerword: Shield, Smite, SW:P, Smite, and Smite some more
8: As above but if your shield drops cast a renew on you to stay healthy till you can cast a shield again.
9: Mind Blast, SW:P, Smite on you crazy Smiters.
Next up 10 and talents!