In the world, players are allowed to find or craft most of the items in each recipe while at the same time they are required of the purchase of at least one item, in each recipe, from the tradeskill merchant. Most of these things can sell for quite a bit of coin on the vendor so you'll really want to go out and get your own materials. Here is a run down of what you will need to craft and some of the better places to find it.
Alchemist: Herbs, plants, extracts, etc are often found on humans. They will drop in caches which you open from your inventory to transfer into your resource pack. Animal items will be found on the corresponding animals (i.e. feral blood comes from wolves) but you can also get some those from the caches. A great tip - visit the Villas in The Noble District. The caches drop with frequency here and the loot will scale to your level!
Architect: All the plan ingredients are purchased from a tradeskill merchant in the resource zones.  They are silly expensive but you don't have to do all the harvesting legwork! You do need harvested items for the construction pieces, and those all come from harvest nodes.
Armorsmith: Your metals, stones, and cotton come from harvest nodes. Silks are human mob drops and leathers are animal drops. A good tip for tier 1 leather harvesting is to stick with the "prey" mobs.  You can kill those really quick with ranged combat and harvest much quicker!
Gemcutter: All your gems, uncut and pristine, will drop from human mobs. Some are rarer than others so never toss a gem out when you get one, even if you don't need it at the moment.
Weaponsmith: The metals, stone, and woods are all from harvest nodes. All of these are pretty easy to gather in the resource zones.
Resource Zones versus Adventuring Zones
If you spend a lot of time out and about in Hyboria you may find that you have little use for the resource zones. Nearly everything you could possible need can be found in the adventuring zones, however, for those who are more focused on crafting the resource zones may be a better choice.
Resource zones are top notch when it comes to node harvesting. The concentration of nodes makes it easy to make your way around the zone quickly harvesting as you go. By the time you get back to where you started, nodes have refreshed and are ready to utilize again!
Adventuring zones end up being the better pick for humanoid and animal kills, particularly if you're already out adventuring. 
You can use the mobs in the resource zones for mob dropped materials but it can limit how much and how quickly you can kill when you are dependent upon a small camp rather than having an entire adventuring zone to play with. 
The upside of using the resource zones for these is you don't have to deal with a full zone of pesky aggro-happy mobs; you can just grab what you need.