The list is ordered in recommended "get" order.

You can check pre-requisites of quests in your quest tree by clicking on a quest you haven't unlocked yet through your Journal.

It also includes the painting names and painters they are based on. Have fun!

• Unlock the Cultural Center and meet Miss Curator by fixing the large building in town
  
• Complete the quests Charity Work and the main story quest Follow the Sign
  
• Catching a Glimpse will now be available for you to get from the quest board and is a pre-requisite for a lot of other painting quests!

Beethoven with the Manuscript of the Missa Solemnis - Joseph Karl Stieler

• Check the painting behind the phonograph in the Music Room inside the Cultural Center

Lady with an Ermine - Leonardo da Vinci

• Complete the quest Charity Work and Follow the Sign

Girl with a Pearl Earring - Johannes Vermeer

• Complete the quest Doppelganger after getting Child with a Carp painting and finishing the quests Dyeing to Know, and Lone Wolf I

A custom painting for a Kickstarter Backer

• Interact with the skull in the Desert Canyon’s far north western cliff

American Gothic - Grant Wood

• Complete the quests Catching a Glimpse, Digging Through the Archives, and Intro to the Rancher’s Guild

Five Bulls - Han Huang

• Have completed the quests Brush Time! and Catching a Glimpse
  
• Go into the barn

Still Life with Cat and Fish - Jean Baptiste Siméon Chardin

• Walk underneath the central large tree in the Mushroom Forest

The Last Supper - Leonardo da Vinci

• Finish the quests Frame of Time, Restore the Mushroom Forest Obelisk, and Precipice of the Moon
  
• You can find it by the entrance of the cave area when you first enter the Frozen Tundra

The Harvesters - Pieter Bruegel the Elder

• Complete the quests Building Materials III, Simple Request, and Digging Through the Archives. Complete Fishing with Nobu at least once
  
• This will unlock the quest Futures Market. Complete this quest to unlock the quest Remembrance to get the painting

Twin Pines, Level Distance - Zhao Mengfu

• Finish the mines questline until From the Depths IV and get The Harvesters painting