Myst Erik Ery's AF/Event guide is pretty helpful for figuring out how to get specific artifacts.
Here's the events you need to do as prerequisites to getting the workshops:
The smithy requires completing the Mine Your Own Business event, which happens at the Ulkan Mines (AF Bottled Spirit).
The golem lab requires completing the Professor Bomb's Lab event, which is started in the Junkyard (AF Broken Doll) by talking to Prof. Bomb, and is finished in Lake Kilma (AF Stone Eye).
The instrument room requires completing the Reach For The Stars event, which happens at Duma Desert (AF Sand Rose).
So, at bare minimum, you need AF Bottled Spirit, AF Broken Doll, AF Stone Eye, and AF Sand Rose.
The most direct way of getting those that I saw was:
Go to Domina and start The Lost Princess, getting AF Stone Eye and AF Firefly Lamp as a reward; place both of them.
Go to Lumina and start Faerie's Light, getting AF Trembling Spoon and AF Sand Rose as a reward; place both of them.
Go to Duma Desert and start Reach For The Stars, getting AF Torch of Coral and AF Broken Doll as a reward; place both of them. You can also return home and get the instrument room at this point. You can also get the golem lab at this point by doing a few more events.
Go to Madora Beach and start Summer Lovin', getting AF Rusty Anchor and AF Moon's Mirror as a reward; place both of them.
Go to Polpota Harbor and start Drowned Dreams, getting AF Pirate's Hook and AF Bottled Spirit as a reward; place both of them. You now have all the artifacts you need.
So, you can get all the needed artifacts in just five events, and then it would be another two events on top of that for prerequisites. Hope that helps. :)
Atheism - A non-prophet organization. "The difference between genius and stupidity is that genius has limits" - Albert Einstein
Corral: Available after you do Little Sorcerers (which requires you do either Niccolo's Business Unusual *or* Lost Princess). Just swing by the back of Domina.
Orchard: Available after you've spoken to Gaeus for any reason. You can do this during Niccolo's Business Unusual, or alternatively wait until you have a *reason* to go see Gaeus (Wisdom of Gaeus or Seven Wisdoms).
- Vilurum Xaren, leader of the happy turtles, and knowitall. ... And if you can't spell "Vilurum", call me "Vil". :)