Assuming I haven't done a single important quest, what are some "must-do" quests that I should take care of before I start doing the useless ones solely for the QP?

All quests are a must do.

Here's just a quick list I made by going through the quest list. Not all must-do's, but recommended do's.

 

 

 

As a First Resort

 

Desert Treasure

 

Dream Mentor

 

Druidic Ritual

 

Fairy Tale Part 1

 

Fairy Tale Part 2

 

Fremennik Trials

 

Fremennik Isles

 

Haunted Mine

 

Heroes Quest

 

The Holy Grail

 

Horror From the Deep

 

King's Ransom

 

Legends Quest

 

Lost City

 

Lunar Diplomacy

 

Monkey Madness

 

Priest in Peril

 

Roving Elves (and subsequent quests)

 

Royal Trouble

 

Shilo Village Quest

 

Smoking Kills

 

Swan Song

 

Tears of Guthix

 

Throne of Miscellania

 

Wolf Whistle