A few motto's from units I served in were, "Ready to Strike", "Tropic Lightning", "Golden Dragons", "Duty First", and "Mighty Proud" (always cringed at that last one).
One of the Gurps military books has a list of military slang, some new, some based on modern ones.
Here are some modern (maybe not so modern now) slang I can think of from US Army:
REMF - Rear Echelon Mother....
Leg - Non-Air Assault or Airborne qualified.
Short - Close to ETS, leaving the service.
Chairborne -A play on "Airborne" for a person with a desk job (derogatory).
PoG - Pronounced poge. For "People Other then Grunts".
Lifer - Someone who is planning on making the military a career.
Oscar Mike or Charlie Mike - Military phoenetic alphabet for "On Mission", or "Continue Mission". Means you are back on mission.
RPG - Not just Rocket Propelled Grenade, also Rape Prevention Glasses for the military issue, large, ugly, plastic framed glasses.
Beans and Bullets - Military supplies.
Barracks Rat - Soldier that never goes out to have fun, stays in the barracks, loner.
The Field - Any area where you are living without regular quarters, camping.
Woobie - Sleeping kit in the field (sleeping bags usually left behind)
Pogie Bait - Extra junk food you sneak with you into the field.
Blood wings - Getting your new Airborne or Air Assault "wings" (a uniform medal) punched into your chest from prior graduates (it pokes the two posts on the back into your chest).
Walking the Gauntlet - Similar to blood wings, but after a promotion, walking down a double line of people with more rank, getting punched in both shoulders (where the new rank is worn).
Dinosaur Country - Rough jungle terrain.
Boot - Brand new soldier (just in from Boot Camp).
Butter Bar - Usually derogatory for a brand new Lieutenant (gold colored bar shaped rank).
Full Bird - A Colonel. "Full" because not a Lt. Colonel, "Bird" because the rank symbol includes a bird.
Smoked - Tired, exhausted.
Jar Head - A Marine (from Army perspective).
Grunt - Infantryman.
Hooah - Can mean about anything depending on how you say it, but most often used as an acknowledgement. A cheerful Hooah can mean you are motivated, an uncheerful one can mean you aren't, for example. Officialy meant to show your esprit de corps, a.i. "Hooah!" after receiving an order (I found it hilarious when I deployed to Thailand and found it also meant "puke").