Veggies are OK as a treat, but they aren't a necessary diet component.
Meats I feed include: goat, deer, beef, lamb, rabbit, chicken, turkey, duck, and cornish hen. Most often available for us are chicken, duck, rabbit, beef, and goat. I should be getting a couple of free whole turkeys soon, so they'll be on the menu for awhile. You can feed eggs as well. My dog usually won't eat the shell unless its crushed and mixed with other food. Since he gets enough bone, I don't care if he doesn't eat the shell.
As for what cuts of meat to feed... I offer everything from tongue to tail. There are a couple things he's picky about, but he'll eat about 90+ percent of a whole carcass (depending on the animal species). I don't grind or mince anything; whole pieces help keep the teeth clean and its easier to prepare the food. Of course, if I have ground meat for myself and don't eat it while it's still appealing, it goes to the dog.
I usually feed liver once a week rather than giving a bit with each meal, but it depends on what I have available. Some people do a weekly meal of liver, others feed a little bit of liver over the course of the week (at each meal). The only supplement I occasionally add is some salmon oil (poured over his meal). If you have grass-fed meat, then you don't need to add oil since it naturally occurs in the meat (feedlot/grain-fed animals don't have the extra omegas in their meat since its not in their diet).