Embark with a good soldier (f.ex. 4 axedwarf, 2 dodger, 4 teacher).
First migrant wave: assign one or two newbies (they'd better be unskilled, more on that later) to his squad and set the squad to Active/Training.
At first, your teacher will lead demonstrations - after some (short) time the newbies have become skilled enough to spar.
Now its a piece of cake, they'll gain skill very fast. If you want: set up some upright speartraps with one wooden training spear (low quality!) each in the baracks, link them to a lever, let your legendary soaper migrant use the lever on and on => armor user skill skyrockets. But make sure you equip your guys with proper armor (bronze, steel).
As soon as more migrants arrive and you feel the need to strengthen your military, split up your initial (now experienced) military and let them be the drill sergeants on the other squads.
For starters, dont make the squads too big. Having too many different kinds of soldiers in one squad slows training down. Make squads of 3-5 and let them gain some skills.
Depending on who has the most experience in a particular skill, that one will lead the demonstrations. This can be a pain if you have f.ex. a talented sword dwarf in your axe squad - he will insist on teaching the others sword fighting. Thats why you should put unskilled newbies together with a good drill sergeant and the more experienced fighters in a different squad. This will speed up the training tremendously, as only the skills the sarge has (and that are the skills you want them to learn) will be trained.
Archers are a bit different, I reckon. While giving a marksdwarf squad a good teacher certainly helps, they gain skill better by actually killing stuff. Since they shouldnt be in the first line anyway (unless they once again forgot to bring bolts -- "Oh you need to bring these little pointy sticks along with this weirdly formed hammer? Wish you had told me earlier, chief!"), training them armor user/dodger isnt THAT important.
And dont miss around with training weapons for their actual training. Never had any injuries during sparring.