Which artifact stats do you aim for?

I have found the artifacts that decrease corruption losses quite useful if your country has multiple towns.

I also like the stats that provide increased weapons damage and those that provide additional experience for fighting monsters

It is very circumstantial, I change gear based on what I am trying to accomplish at the moment or who I am attacking/defending against. For fighting monsters the +creature, + close-in (unless you are fighting with a bow) are very helpful and + land for appropriate monsters; +aerial help with dragons. If defending anything that adds health to hero and health/armor to buildings. When attacking/defending hero health is important to keep your gear/perks working. Army attack, +charges, +unit energy, + healing are also good. It is best if you know the kind of units your opponent has and can select your gear based on that, for example, if there are a lot of greek fire +wheeled is good, if there are a lot ships +attack on naval would be good. If you are trying to level up your hero anything that is +experience.

Onslaught: increase number of charges + energy + army attack + army speed

Monsters : chance to pick 2 items instead of 1 + chance to pick up research points + chance to pick consomables + increase damage given by hero against creatures

In town : increase all ressources production + increase time construction + decrease coruption for production ressources

Like Kmcy already mentioned changing armor depends on what you are doing.I'm using it a lot like her.I also try to enhance as much as i can.And visit the idols often.Thanks for that tweak btw.Much appreciated.

I usually go for ones that provide skill boosts. Those with certain % of corruption mitigation come second in my priority list.

Which skills?

Bonus skills like +x wisdom & intellect from artifacts have helped me with the riddler’s questions as high enough level of those traits highlights the correct option in provided list of answers. By making use of proper artifacts, I don’t have to get the required stats solely by spending skill points. Artifacts with bonus charisma might be helpful for quests that require certain base level of charisma. I also like that these artifacts add to our army stats, however little it might be. A reasonable boost can be achieved by using multiple artifacts with similar stats.

I always look for those that boost army health + army attack (both directly and indirectly via +charisma/intellect). It's good for all fighting. Next in priorities is hero healing points (for mobs) and unit charges (for siege attacks). For mobs I also like to have 2 items instead of 1 and some % to science points. When settling obviously I dress up all cultural - I have found that usually the corruption decreasing items tend to result in a bigger increase in culture than simple +culture items.

Btw, I have NEVER managed to improve an aritfact so that it is more useful (even though I have the enhancement perks maxed). It always gives some useless stats. Sometimes I used up like 200 crystals and it got worse and worse... At the end becoming totally useless with hero lev needed increased to 50. Now I only use crystals for repairs (and I feel there's definietelyy not enough of them if you want to play intensively - even if I increase consumable drop chance, the number of mobs I need to kill to get some of the rarer crystals does not cover the wear out done by that fighting).

I think the perk that increases enhancement chance should give you some chance of "steering" the enhancement towards some stats you like. Otherwise, you're much better off increasing equippable find chance + artifact find chance - you get better items this way than through enhancing.

And while I'm at the idols topic I have to say crafting / disassembly proved even more useless for me - if by disassembly I can get 1 ore with some % chance and then I need 5 ores to create the SAME class of item with again only % chance then seriously, it's useless. Not worth investing all these perks - much easier to invest in item finding. Why not change it so that while disassembling you get some % chance of crafting a random item of the same class? And if not lucky - an ore? Or maybe disassembly should give you either 5 (maybe 4) ores of the class of the item or less ores (1-3) of a better class? So that you could get rid off many weak items in the hopes of getting a better one rather than a hope of getting exactly the same of which you have just destroyed 20 or 30... Right now I find it easier to throw items to trash rather then disassembly them - much faster this way, the result being the same. (Btw, when you need to disassembly or repair items/item repeatedly the idol animations turn from cute to boring to downward annoying - a button to skip them would be nice...)

I like to wear the ones that speed up build rates, fishing/or treasure hunting speed when not fighting. While fighting I like the ones that boost the army health and allow me to pick up 2 instead of 1 option, and speed up army movement. I do not like it when I forget to change my outfit though.  

increase science points, culture, resources, things that give hero armor and health, experience from quests and fighting monsters, army attack, research losses due to corruption