My goal (sort of) at this point is to push back into the population a larger group of dwarves with a military skill, Armor and Weapon smith at around Level 4, and possibly planting or another Farming skill at 3 or 4. A smaller group will be the "industrialists" with a higher skill level in 3-4 other areas.
Generating a high-level specialist is what happens a lot --- and then you never get the specialist you want as a migrant. Thus a reason to spend the time getting a less military skilled dwarf good at a number of skills so that a fort will have a slot he fits.
I saw the utility of this when a dwarf arrived in the 2nd wave at my current fort who was level 10+ in 4-5 skills (Weaving, Clothier, plus Bone Crafting, as well as 4-5 in the main smithing skills.) He essentially was better than any current dwarf in the fort in that entire skill set. And with his work speed he could cover a lot of ground and basically proved very valuable to a young fort.
Whether a fort makes a good training environment for which skills will also depend on the fort's resources and industries:
I had a lot of shallow tetrahedrite veins plus the Magma Sea was only ~26 levels down. Therefore the Foundry was established early and there was a lot of copper bars to play with. Easy to queue up cheap armor and bolts as smithing training and ammo for the marksdwarves. Black smithing and metal smithing can be trained as well with buckets or something else that can be melted down to recover the metal. (Which trains Furnace Operators of course.) And, as a side effect, these are tasks that help with increasing Strength, Endurance, and Toughness.
Your farming/livestock industry will generate a lot of training tasks: Miller, Thresher, Planter, etc. and then Butcher, Tanner, Weaver, etc. in the follow-on processing industries. And if you have a silk farm running training Weaver is even easier. None of these are "vital", but are useful for a fort to have someone show up trained in if that is an industry the fort is trying to set up.
The military skills is essentially being patient, or using a danger room. I tend to draft most of my workforce at least part-time into training by the 3rd year. Get them doing combat drills or archery range with their idle time rather than partying. And once they get a level or two in a basic weapon possibly grouping them more efficiently to see if they will spar. (I also use the draft to get them uniformly clothed, using some armor that won't wear out, and another way to get them physical training.)
Haven't hit upon a good method for getting gem cutter/gem setter trained. It would probably gear around having a glass industry generating lots of rough green glass and having them cutting and setting that as practice. Though they can also just work with stone as well. (Which makes it easier than in previous versions.) That would depend on how your stone situation is going - in the current fort mine went into blocks for paving, surface construction, and mechanisms. Glass and clay took up a lot of the workload in terms of furniture, food storage, and building as well (clay brick). So getting trained potters and glassmakers was not a challenge once immediate needs were met.