My order of preferences for artifacts is something like:
S tierThese should be your top priorities, and you should be willing to sacrifice anything else below to achieve those.
- Legendary Alloy Values: This is the top priority of everything since it allows you to do market strategy effectively. Don't make it insane yet.
- Rare Hospital Boost: Save armies; it is fairly easy to get. Don't upgrade over rare since it only provides marginal benefits.
- Insane Army Cache Boost: Who doesn't like lots of free armies?
A+ tierVery useful, always try to keep those around. These are also prime candidates to upgrade to insane after S-tier is completed.
- Army Camp Boost: Very helpful during the early stages so you can get to the markets faster.
- Cache Boost: A worse version of army cache boost.
A tierUseful, but not as good as A+ as those are more situational, unlike A+ tier artifacts that are useful throughout the whole game.
- Supercharge Army Camp: For battles and has synergy with clan requests and free cache powers.
- Time Wrap: For battles and skipping hospital upgrades.
- Mercenary Discount: This saves a decent amount of money on some levels.
B tierA bit less good and even more situational than tier A.
- Resource Cache Boost: More money during early levels and help you get some techs faster.
- Tech Discount: Help you get techs faster.
- Speedy Crafters: Reduce the wait time for crafting on some levels.
- Money Cache Boost: More money, but more useful for later levels, but generally worse than RCB.
C tierI would try to keep those when combining fodders, but the benefits are quite marginal.
- Draft Boost
- Army Camp Discount
- Mine Boost
- Bonus Money Boost
- Territory Money Boost
- Ore Values
- Item Values
- Speedy Smelters
- Hospital Discount
Honorable MentionsThese are quite specific if you want to target Hex and post-super levels.
- Triple/Quadruple Strike
- Inspire Mercenaries
D tierEverything else except...
Meme tierWhy do those even exist?
- Fog Buster
- Field Hospital
- Discount Army Camp/Mine/Hospital