They have two passives, one that is always active like Ifrit's attack bonus and one given during the actual summon. Leveling mainly increases their summon bonus along with giving an extra summon turn once they reach lvl 11.
Artifacts are pretty simple. Fuse a bunch of the same ones (such as 4 Terra ones) to reach full limit break (the gem/materia slot's are filled, also text changes when choosing them), then feed it a few more artifacts to max it's level. Pray and hope you get a good bonus afterwards. The bonus is an equipable cp passive, some are very good. But the current artifact grind is..pretty bad. Also eats up gil pretty quickly, so be warry of that. Gets much better in a future update though.
Probably early on, clear artifact WOI at least for gems, see how far you can get in a summon WOI. Then, if you can farm the boss levels with little trouble, then would try to get Ifrit, Chocobo, Sylph, to lvl 11. They tend to be staples of the Co-Op's and most general content.
