Spirits of the Elid will give you a good amount of Prayer xp - Look up others on Rune Hq
the best way to get your prayer to 37 is to do quest's that either reward xp lamps or actual prayer xp. Or if you are willing to spend money buy dragon bones and bury them, they give you 75xp each bone, or the best way is to buy lots of big bones. another good way to get prayer up is to go to pest control and buy prayer xp with your points.
Babydragon bones give 30 prayer xp per bone if you bury it.
there actually is no best attack xp quest for runescape but you could try some members ones
Well, any quest that involved a specific skill give you that experience point.
Temple of Ikov give a decent amount of ranging experience.
you can level up by doing quest and fighting boss which give you many xp points
keep battling high level monster and keep doing quest that give you a lot on xp
the restless ghost is the only free-to-play quist to give prayer exp i think it gives 500 exp and lets you change your grave marker instead of that n00bie flag we start with ;]
Quest grinding is just doing all the quests that you come across. This is different that mob grinding in that you get more XP (experience) from doing quests because you get the added XP from the quest AND the xp from killing the mobs.
for a non-member that's at least lvl 48 (to be safe) the dragon slayers quest (and you need 33 quest pts i think).18,000 strenghth xp and 18,000 attack xp.
There is a quest called "The Digsite" .It gives you at least 15,000 exp. Its an intermediate kind of quest and its a lot of fun http://runescape.wikia.com/wiki/The_Dig_Site
You battle beasts.