For good 6* artifacts the best way is still the dungeons at level 20 (Ice Golem's Peak, Dragon's Den, Fire Knights Castle, Spider's Den). 6* artifacts do become available from level 13 on all of those dungeons, but drop (a lot) less frequently

Another way is the Forge (pick the 5*-6* group) and the material for using the forge you get to collect from Tournaments and Events (the Forge Hammer), from classic Arena (the iron bar) and from level 15+ of faction wars (legendary stones)

To add to what OzzLee said, the Ice Golem, Dragon, Fire Knight dungeons each drop certain sets (you can see which ones on the left after clicking on them, in the screen where the stages are listed). Generally Dragon is considered top priority, because it drops the important sets speed, accuracy and lifesteal (plus, it's often easier for people and requires less complicated teambuilding, though that depends on your roster of course). 

While Spider only drops accessories, so rings, amulets and banners - and since Spider is the hardest dungeon for most people, getting 6* accessories is a bit tougher.