This is the first way we're introduced to each character—by what he "carries." For example, Kiowa carries an illustrated New Testament (given to him by his father), his grandmother's distrust of the white man, and his grandfather's hunting hatchet. He wears moccasins for silence. So, from the very beginning of the story, we can guess that Kiowa is a) Christian, or respects his father, b) an American Indian, and c) seriously tough.
Henry Dobbins carries a machine gun and his girlfriend's stockings around his neck for luck, which tells us that he looks scary, but he's just a big old softy, really.
Rat carries a medical kit, comic books, and M&Ms for really bad wounds. This shows us that he gives chocolate to the dying, which means that he doesn't ignore the fact that the dying are people just because they're dying. That makes us like him almost immediately.
In "Spin," O'Brien reintroduces us to the soldiers we already met in "The Things They Carried," plus a few we haven't met, and this time he uses actions to show us what they're like. Ted Lavender, for example, adopts a puppy. Then Azar straps it to a mine and blows it up. This means that Ted Lavender is a nice guy, even if a little impractical (who adopts a puppy in a combat zone?), and Azar is a jerk who's potentially also a sociopath.
Norman Bowker whispers in the night about how he wishes his dad didn't care about medals, so he's both ashamed of looking like a coward to his father and ashamed of the fact that he's ashamed in the first place (or he wouldn't be whispering).
Kiowa teaches a rain dance to Rat and Dave Jensen, showing that Kiowa's willing to teach it, and Rat and Jensen are willing to learn it.
For soldiers, occupation is really about rank and field specialty, and this doesn't work for all the characters because most of them are just common grunts. For some of them, though, it's pretty interesting. Rat, for instance, is the medic. If you check "Character Roles," you'll see that he's also the storyteller. In this book, stories are all about healing memory and raising the dead, so it fits perfectly with his job as medic.
Mitchell Sanders is the RTO, or Radio Telephone Officer, and that fits in with his role as the one who's always trying to hear and communicate the truth. Finally, Jimmy Cross is the officer, which fits his role as the one who's obsessed with responsibility (though, in his case, this means blame).