Huge edit: THis is for 31.10.
I use the military. I am really not having such big problems. At the 2nd immigration wave (the one that brings you up to ~40 pop) I usually start training 10 with iron axes. This is a short guide to off-duty training.
TL:DR; don't touch the whole on-duty/inactive part of the military screen. Dwarves will train fine on their own, using individual combat drills.
Steps to produce a working military.
1. Mentally mark 10 dwarves to be in the military, either through nicknames or profession (for example, Military1). This makes it easier to assign them later on. Make sure noone has any labour assigned to them.
2. Make a new uniform in the military screen. Make it 1 weapon only (no armor yet).
3. Make a new military squad. Select the uniform you made, as you create the squad.
4. Leave the squad inactive
5. Build an armor stand or a weapon rack (1 is sufficient). Make it an armory. Q over the armory, and select your squad to Train there. Each squad needs an armory.
6. Dwarves with nothing to do will now go for personal combat drilling. However, not all dwarves are cut out to do this. Usually its about 8 out of every 10 that will take the initiative to do individual combat drills. Remove the others from the squad. Replace with new ones. Note that they do not instantly gain the idea to go training - however, if, after a couple of minutes, you can find them at one of your meeting places (jail, dining room, zoo, statue garden) then they probably will never train on their own.
That's it. You've now got them training. But what about when we want to upgrade their equipment?
7. (F)ree up the armory. Disband the squad. Make a new uniform with all the right equipment. Make a new squad. Assign dwarves. Assign squad to armory.
Be warned. Dwarves that do individual combat drills when off-duty and nothing else to do are fiercely loyal to their chosen pastime profession. They'll spend almost all their time training, which means they'll have very little spare time to witness the wonders of your fortress and they are therefore missing out on lots of happy thoughts.
You should also be aware that individual drills only train their main weapon, and dodge to about 1/6th of the efficiency of their weapon skill. Therefore, if you want to train shield use, you'll have to make the squad with a shield and without any weapon.
Pro-tip: Build the armory outdoors. That way, your dwarves will not get any negative thoughts for sunlight.
When the time for action comes, simply send your military towards the enemy using the (M)ove command. The dwarves will automatically switch to "on-duty". As soon as the enemy is dead, switch them to off-duty by removing their orders. It is very likely that they will suffer several negative thoughts, like "seen death", "was taken off-duty recently". However, this is not a problem. They will grumble, but they won't riot. Watch out for zombies leaking miasma though (lots of extra unhappiness there).
Pro-tip 2: As your dwarves reach legendary +5, which will happen after about 4-5 years (less, depending on how often they are fighting real enemies, and how little they spend in hospital beds) either switch them to train shield use, or take them off individual drills by removing the armory. This will free them up for experiencing all the joys of your fortress, and it will free them up for performing other tasks, such as hauling.
Edit: Also note that dwarves will take newly made gear if its better than what they have. If you are making lots of new gear in a very short time, this can create a loop where one dwarf runs and picks up a piece, placing his old piece in the stockpile. The next dwarf now sees that piece that the previous dwarf left, and goes to pick that up. The next dwarf sees the piece the previous one left. With 8+ armor pieces and weapons for each dwarf, if you have multiple squads, they can literally spend years doing nothing but switching gear. To avoid this, simply disband all squads and free the armory. All freed squads will dump all their items. Now make a new squad, and that squad will take all the best gear. Then the next squad, and so on. This is also an effective way to make sure all your masterwork gear ends up in one squad.
Pro-tip 3: I usually start my 2nd squad out with shields first, thinking long-term. A legendary +5 shielddwarf will block every attack made against him from the front. After the shieldtraining is done, I assign weapons to them. Note that this squad will most likely never see combat until they are shield/weapon legendary +, as they are too precious to waste, and deaths still happen.
TL:DR; don't touch the whole on-duty/inactive part of the military screen. Dwarves will train fine on their own, using individual combat drills.