It's not as scary as it seems. The key concepts are:
Squads\Squad Leaders
Alert Status
Schedule Status
Equipment
You choose a squad leader and create your first squad. It asks for an armor type (Metal Armor) which by default lets the dwarf choose the best weapon and armor he can think of. Equipment settings are complicated to deal with and do have some bugs, but the default "Metal Armor" setting has served me just fine so far.
Once you've created a squad, you only need to do one more thing to make them operate. Get out of the military screen and press 's' to bring up the squad interface. You'll see a bunch of confusing shit. Press 'a' to select your first squad. Press 't' to change this squad's schedule status. It should now show as "Active\Training" - your dwarves are now active and in the military. If you have a barracks, and you've enabled (T)raining at its building, they will now practice there.
Now that your military is active, leave them that way. They receive a significant unhappy thought from being set back to "Inactive", so if you have enough dwarves to get all your work done, I recommend a permanent military. However, once set to "Inactive" all your dwarves should be capable of being given civilian jobs again.
To actually order your military around, there are 2 major easy options. One is to use 's', 'a', 'm' and then press enter--this issues a movement order. Your soldiers will go to that location and stay there, fighting anything they feel is appropriate to fight. The other option is to use 's', 'a', 'k' and then select an enemy unit and press enter--this issues a kill order. Kill orders can be buggy and sometimes don't work, so in general I'd recommend using move orders instead.
When your military is done dealing with threats via either method, you must remember to cancel their active orders by opening the military interface and pressing "c". This will delete the active order and let them return to training, eating, etc. If you forget to do this, your soldiers will dutifully obey you unto the point of going berserk, melancholy, or dying of hunger\thirst.
There is a lot more stuff that can be customized, to set the size of active squad staffing at various guard posts, patrols, uniforms, and much more. But until it all works a bit more fluidly, I'd recommend doing as I have done--learn to make them function in the basic ways that are necessary to protect a fort, an otherwise don't fuck with it.