Military in DF is super easy but people get into trouble when they try to make it more complicated than it actually is.
Here's my foolproof strategy:
When the first migrant wave arrives, choose the 2 best soldiers and put them in a squad of 2, training constantly non-stop, with wooden shields and training weapons.
Same for each new migrant wave.
When the caravan arrives, buy some metal armor and weapons for your 6 soldiers (3 squads of 2). Switch the squads to "metal armor" uniform. If you have metal on-site and can forge your own, even better, but not strictly necessary.
If you stick to the plan you will have 6 soldiers per year of your fort, and they will be Legendary in their weapon skill in no time at all. The trick then is to put their training barracks at places where they will be useful, like your main entrance, your cavern entrances, the trade depot, etc.
For marksdwarves, I train them as hunters, so they will become Legendary while simultaneously providing the fortress with huge amounts of food, which I cook into lavish meals to trade for all the weapons/armor/ammo/bars/anvils I need.
I was under the impression that making 100% of your squad train 24/7 with no relief or rotation really pissed them off. I've had dwarfs' happiness tank because of "patrol duty" even if they're only ever set to train, and that was in a squad where 8 of 10 were training. I know if their skill gets high enough they upgrade to their 2nd tier unit and this ceases to become a problem, but I can rarely get most to that point (especially melee) before they run out of happiness and need some off-duty time. What am I doing wrong?
I've never had a problem with military happiness; in fact they are often Ecstatic (took joy in slaughter, etc.) while my civilian dwarfs are miserable.

Training them 24/7 has 2 main benefits: first, they train faster (obviously), second, they do not mingle with civilians, which means they do not form friendships creating bad thoughts if one of them gets killed.
Needless to say you want a door/drawbridge between your barracks and main fortress, so you can isolate a berserk legendary swordsdwarf from the general population if necessary.