When should I start collecting Victory Points in Dominion?
This question is really dependent on the Kingdom cards available and no overreaching general rule can be made without a lot of major caveats.

One good strategy would be


Always buy a province when you can afford it
(Unless you don't have a gold yet, and the game is still young)
Start buying Duchies once one pile is empty if you think the game will end by 3 piles disappearing
Start buying Duchies once half the provinces are gone if you think the game will end by province depletion.
Start buying Estates when you think you have at most one shuffle left in the game.


Again, this is general advice and specific cards can change things up drastically!