Essential workshops are (imo this order) Carpenter, Mason, Mechanic, Still, Craftdwarves, Smelter, Metalsmiths Forge. Much seem to unnecessary. I'm not sure if you need a loom or Clothiers shop in the current version. I have a HUGE pile of quality new silk clothes, but still all of my dwarves are running around in very worn elk leather.
This depends all really on where you start with and what you brought with you. Your order of workshops wouldn't work for me at all.
First and utmost important: you need a farming plot. Since subterranean ones require irrigation activities, start in a temperate lush forest type of area and collect some strawberries (or similar) and plant their seeds. The dwarves can eat those while you're discovering the joys of subterranean farming and mud creation.
Build a
Still so that they can convert these aboveground plants into booze. Booze is absolutely vital. Much more vital than fresh water or something like that.
To secure your pile of goods from the wagon, you want to dig a moat of some type and to defend yourself with traps. This is where the
Mechanist becomes important to have. Build some mechanisms and use them while you're still confused about the subterranean farming. You'll most likely need 3 mechanisms anyway if you're using a cistern/water source + door/floodgate setup to create the mud for the farming plot.
Speaking of doors and floodgates, you need the
Mason for these. He can also build two tables and two thrones, so that you can set up a little office for your freshly appointed manager dwarf and as a pityful little dining room so that your dwarves can sit down to eat those strawberries. Expand this later. Dwarves like a good dining room.
The mason or anyone else should also be able to do
Architecture, because the best defense is still a raised drawbridge at the sole entrance of your fortress. Bridges require a bit of architecture to be built.
Since you're in a lush forest, chop down some trees and make more barrels in the
Carpenter's workshop that you only need to build now. If you don't have any trees on the map, you won't be really needing the carpenter's workshop for quite some time, except for the production of a bed at some point. The carpenter is useful (if you have trees), but not vital.
At some soon point you'll discover that you need bags. Rope reeds and Pig tails are plants from which you can produce cloth bags from; there's an abundance of silk in the (dangerous) caverns; you buy stuff from caravans and get bags from them; or you can butcher down some animals and make bags from their skins. Either way, this is going to be a bit more complex with another farming plot, a
Farmer's Workshop,
Loom,
Clothier or
Butcher,
Tanner,
Leather Workshop. Or a
Trade Depot and something to produce export goods with.
An absurdly valuable source of goods for the caravans are
Kitchens, by the way. Lavish meals can easily be worth in the thousands of dwarfbucks, if they contain some processed products of slightly higher quality standards.
Craftsdwarves, Smelter, Metalsmith... that's absolutely not vital at all. You can just sell your lavish meals in the start instead of having a potentially useful dwarf sitting all day in the craftshop, producing rock toys; and while armor and weapons are good to have indeed, you can defend your fortress in the beginning just with traps and drawbridges, especially given that the military screen is just too confusing for newbies anyway.