To save you a lot of time and frustration: forget about sparring. It won't happen. At least not yet.
Do not use the Active/Training alert. The only thing you should be using alerts for is a "Danger" state wherein your dwarves cannot venture outside (by being restricted to a burrow that encompasses only the insides of your fortress) in times of sieges or ambushes. Fear not, however: your military dwarves do train, they just don't spar. Setting them to inactive and leaving them alone will have them spending most of their time in the barracks doing individual combat drills, which are solitary activities that passively increase combat skill. This is provided, however, that you have your barracks designated to be trained at by that particular squad: through the q menu, go over to your barracks. You should see any and all squads you have in the panel, with the options train, sleep, etc. Through these options you can toggle what squads do in this barracks (if you don't do this, they won't recognize it as a barracks at all). Ideally, make two barracks. One full of beds, and one simply made from an armor stand or weapon rack with a fair amount of room. In the first, have your squads sleep; in the second, have you squads train -- through the menu mentioned above.
Provided you do this, you'll see your soldiers hanging out in your barracks. Also, forget about crossbow-dwarves; they don't work yet. And once an arsenal dwarf becomes available in the nobles screen (when your fortress population's high enough), appoint one! Without it, your military can't function because now their equipment is assigned and managed by said arsenal dwarf.
Hope this helps. :3