Every dwarf gets conscripted for at least baseline military skills so they're not utterly worthless when the shit hits the fan. I have a bunch of squads to organize my dwarves and to keep my important ones notably safe and out of combat.
My civilian dwarves are assigned to a few special squads - 123THEDOCS is my medical team, any dwarf with any medical skills gets put in here, above any of the other special squads. The Stone Mothers are the baby-carriers, married to the Iron Fathers who become my captain-of-the-guard's squad, the innermost fortress elite guard. If there are more married pairs than 10 these squads graduate to Granite/Steel, and Slade/Adamant if the fortress were to ever get so large. Creative Carnage is my legendary crafter squad, with minimal training time because they gotta produce the top-quality goods and are usually doing so. Swarthy Scribes are the library crew, and occasionally the farmers get thrown in there as well.
After all of the special dwarves are so assigned, I can more easily assign the other dwarves without worrying about accidentally putting the 6th Vabok who is my new legendary Blacksmith into the frontline squad. Marksdwarf squads are named the First/second/etc. Rangers, while melee squads get First/second/etc. Blades/Carvers/Lancers/Beaters/Crushers for their respective weapons of choice and proficiency. If I'm lucky enough to get a dwarf with a very high Teacher skill and some military skills, he becomes the captain of Fight School, where the fresh meatrecruits train until they actually have some skills. The Battle-Scarred is the injured roster.
Most dwarves are placed into their squads once, and never move. Some dwarves will get moved into Mother/Father squads, a crafting squad, or the injured roster. If I have training squads (which I've only had a fort exist long enough to make use of once) then they'll be placed there before finding their rightful permanent squad.