The problem is traditional training (i.e. no danger room) is very slow.
Having small squads on active training obviously helps a ton, although I would advice having 3 dorf squads, setting up standing orders to have 2 dorf per squad training all the time, so that one dorf is free to go to sleep every so often. You can also set-up dedicated barracks (beds), dining hall and small drinks/food stockpiles near your training area to minimize time lost during breaks.
Also, the squad leader will very slowly gain teacher experience, I don't think you can make this process any quicker.
A good way to bootstrap the whole process is to rig a dwarf or two among the initial seven to be dedicated teachers, i.e. proficient teachers + points in whatever skills you want your military to quickly learn (a weapon skill + dodge are a good choice).
The kind of weapon skill your immigrants come with is very luck based. If you find yourself with an overabundance of dorfs with an unwanted weapon skill, feel free to regig them into another weapon (you will have to set the other desired weapon type specifically in the military interface, as leaving the setting on personal preference will result with them in picking up the weapon they are most skilled in, if that's available to them).
Keep in mind, you can just draft more unskilled paesant to be trained in using the weapons you're looking for, while leaving the unwanted macedwarves/whatever as replacement for those who fall in battle.
I usually switch to larger squads once I have a few legendaries, or whenever my military grows too much and having multiple 3-dorf squads is too much of a mess. Be sure to assign your most skilled dorfs as squad leaders, as I think that helps with making the teaching process go more quickly.