This table will show you the basic quests you should do before training

Tutorial Island
Cook's Assistant
X Marks the Spot
The Restless Ghost
Rune Mysteries
Imp Catcher
Witch's Potion
Client of Kourend
Dwarf Cannon
Waterfall Quest
Tree Gnome Village
Monk's Friend
Hazeel Cult
Murder Mystery
Merlin's Crystal
Holy Grail
Druidic Ritual
Witch's House
Black Knights' Fortress
Recruitment Drive
Observatory Quest
Priest in Peril
Rag and Bone Man
Nature Spirit
Alfred Grimhand's Barcrawl (miniquest)
Scorpion Catcher
Plague City
Fight Arena
Gertrude's Cat
Natural history quiz (miniquest)
Jungle Potion
Vampire Slayer
Death Plateau
Goblin Diplomacy
The Queen of Thieves