Depending on the set, first I fodder flowers and plumes:

1. If it has 4 stats out of which 3 are good then I'll keep.
2. If it has 3 stats out of which 2 are good then I'll keep.

Pay attention that healers' sets such as Maiden Beloved or defenders' sets such as Husk of Opulent Dreams prefer entirely different stats.

Then I fodder circlets/sands. If main stat is good (set dependent but usually CV/ATK/ER/EM) then according the flower/plume procedure. The exception I make is to not fodder EM or Healing Bonus artis since those are generally rare.

Last are the goblets. Same rules. Might not rush to fodder Elemental Damage bonus ones as well since those are rare too.

I keep 4* artis only if they don't have a 5* version, plus have desired main stat and three sub-stats all of which are good.

Personally I only use the strongbox at the point where I have no more artis to fodder into according to the above but that's just me.