So if you're like me you never gave artifacts much thought aside from making sure to keep a few shells and cut gems around so you don't lose important dwarves, but why leave things to chance? There's a better way!
The first thing to keep in mind about artifacts is the "highest moodable skill". Whenever a dwarf has a strange mood what kind of artifact he makes (or attempts to make) is determined by his highest "moodable" skill (if he has no skill in any of the moodable skills he make a generic "craft" type artifact, not good).
Exactly what item is made is picked more-or-less randomly from the list of all items that can be made with the skill in question, but there is one exception: if a dwarf likes one of the objects on the eligible list he will (almost) always make that item.
Good skills:
Blacksmith is really good, as you will get a piece of furniture made out of valuable metal
Mason and Miner will give you a piece of stone furniture, also really good particularly if you force the use of valuable metal ores for the stone.
Weaponsmith gives you an artifact weapon. Awesome!
OK skills:
Carpenter gives you wood furniture, not as good as the others but still better than a craft.
Mechanic gives you an artifact mechanism, the usefulness of this is determined by if you feel the need for a really good weapon trap or high value well.
Armorsmith has far to much of a chance to get one boot or glove, but if you have a guy that likes breastplates or greaves...
Bonecrafter gives you mostly trinkets, but sometimes you can get something useful (like a pick!).
Glassmaker once again gives mostly trinkets, but sometimes you can get some really good stuff.
"Bad" skills:
Bower 66% chance to make something (nearly) useless? I think I'll pass.
Woodcrafter and stonecrafter give you worthless trinkets, yick.
Clothier gives you cloth items, and who needs an artifact shirt or bag?
Engraver just gives you a craft, not worth it.
Note that depending on how you structure your industry it may be worth training otherwise "bad" skills just for the legendary dwarf. This is mostly a consideration for skills that are hard to train like armorsmith and engraver.
Once you decide what kind of artifacts you want to make start training all your "useless" dwarves in those skills, you should only train in one skill per dwarf and only the minimum required to make that skill his highest, for most dwarves dabbling is good enough. If you have to train much beyond dabbling, it is probably a good idea to just skip that dwarf.
It is generally a bad idea to train all the dwarves in your fort in one skill, spread things around a little so you get some variety.
The best way to train a skill is to set a repeat job using that skill in a workshop that only allows dabbling workers (set the max skill to dabbling). This should make all dwarves with that skill enabled train until they are novices, at which point they will quit and let the next dwarf train.
Now that you have most of your fort primed to make the "correct" kinds of artifacts there is only one more variable: material.
There are three ways to force artifacts to be made from the most valuable materials:
Use DFHack job-material
Forbid everything but the most valuable stuff
Build a special artifact construction area
Using DFHack is the easy way, but most people (including me) consider this cheating.
Forbidding everything works, and it will be the only way for early artifacts, but doing this will cause everything else in the fort to grind to a halt. As an alternative consider just forbidding things like low value gems and metals. Of course it goes without saying that a bookkeeper makes this much easier
The artifactory is the easiest system for forts past the first year or so, but you have to remember to build it ahead of time :p
To put it simply an artifactory is a large room with small stockpiles of the most valuable materials in the fort and a selection of workshops. When a dwarf gets a mood check his skills to determine what kind of workshop he will need (for example a weaponsmith will need a forge and a mason will need a masons workshop). Now forbid all of the workshops of that type in your fort EXCEPT the one in the artifactory. The dwarf will go to the artifactory and claim the workshop there (if he starts to head towards a different workshop then you misidentified, try again). As soon as he enters the artifactory lock the door behind him, now all you need to do is check on him now and again to make sure he has everything he needs. If he gets stuck you can temporarily let him out, but as soon as he drags back whatever he wanted lock the door again to prevent him from fetching low value materials from outside.
When the artifact is finished it is a good idea to forbid everything inside the artifactory so that the buildings and items are not used for non-artifact purposes.
By far the hardest part of making an artifactory is getting enough of everything inside.
Here is a list of approximately what you will need:
4-6 metal bars
4-6 boulders
2-4 blocks
4-6 cut gems
4-6 rough gems
3-4 yarn cloth
3-4 plant cloth
3-4 silk cloth
3-4 leather
2-4 bones
2-4 shells
3-4 wood
1-2 fuel (if artifactory is not magma powered)
The above list is just the basic items, for example if a glassmaker mood is possible in your fort you will need raw glass as well.
In my test fort I have had my share of useless trinkets, but most of the trinkets were made by kids (who can't be trained) and dwarves who had a "bad" skill as their main profession.
Early on I only forced selection of silk, metal, and gems. This worked fine and I got a lot of valuable artifacts out of the deal, but now I wish I had built an artifactory sooner...