Leather armor trains armor user. The reason I suggest leather is mostly a cost thing, I'm usually training two or three squads at a time and when I start I rarely have enough steel or even iron to make them armor. A lot depends on your time table, when you start building a military the first fall you rarely have the infrastructure to do the smelting to make lots of armor, but leatherworking has no overhead except getting the leather, which I usually embark with, it's pretty cheap. I guess dig deeper has conditioned me, against goblins you probably have time to get mid to large scale metalworks going.
However, you don't need heavy armor to train wrestling. Dwarves can train wrestling naked and almost never (I mean, less than .01% chance) kill or even injure each other. In all my forts, I've only had one death caused by a wrestler (and it was a fortress guardsdwarf, ironically). Training armor is just for skill ups, you can worry about real armor later if you don't have it already.
You do want to start training early though. Draft a few useless dwarves out of each immigrant wave until you've reached your goal. Ambushes can start within the first two years or so, possibly earlier (I've seen them with only the starting 7 before), and even wrestlers in leather can take down 8 goblins.
Well, yeah. I guess military to me is optional because you can do as good as a military with traps and constructions. Heck everything's optional if you get down to it.
Yea, everything's optional. I've played forts in dig deeper with no military, just had a giant drowning pool. Worked great until some goblins showed up, but that was my fault, should have had mechanical traps too. The military is fun though, I'd recommend everyone give it a try at least once. It's really awesome to watch your marksdwarves cut down orc squads while your hammerdwarves knock them 40 tiles away into a tree, or to see speardwarves instantly turn enemies into corpses and just move on. It's an interesting infrastructure challenge to get enough steel to make armor early (especially if you're into military challenges, like orcs) and have a dominant military before the enemy even shows his face.
It's also nice to have the flexibility. It takes a lot of traps to protect liaisons or guard the dwarves cleaning up the traps after the last ambush was butchered. I've lost haulers when a second ambush arrived on the heels of the first and attacked before the jammed traps were cleared. With hammerdwarves, I can spread my squads out (I like lots of 4 dwarf squads) to blanket the area. Goblins try to path to the nearest dwarf, in my experience, which will mean a champion, so my haulers can work in relative safety.
OK, the leather makes sense as a cost thing, and I wasn't particularly sure before about the ability to train armor user via leather. The main reason why I suggested starting off with heavy armor was that my own personal experience seems to show minor nerve injuries more with lower levels of armor, and that said, there's no reason why you shouldn't start off newer squads training with full equipment and shields if you can.
As for orcs, my own strategy is to prepare bone/wood crossbows and bolts and snipe them from battlements the first year, while robbing the first dwarf caravan of armor, so cost isn't an issue.

But it does depend on how you want to play the game (as does the OP's question, in a way), and it's interesting to hear how different people deal with the same military situations...
My own strategies for sieges and ambushes are to lure them into killing fields that can be sealed off from the outside with ease, and to use chokepoints so that ambushers are intercepted by the two or three champions I place on duty at the start of each season (which I set to carry food and water). I then wait for a siege to end before I send haulers to collect loot. I'll admit that I haven't got a good solution for liaisons spawning all the way across the map in the path of ambushes, but that problem's also partly related to the size of the site.
Ultimately, once you get past the beginning, there's no hard and fast solution; as everyone else is saying, in the end it's down to your personal playstyle and preferences.