The answer is.............it depends. If I'm close to a road and it's cool, I just open the deer up from the testes to the brisket and I even leave the bladder in until I get the deer drug to the truck. I like to open up the hind quarters and cut the pelvis with a saw to remove the bladder. It's cleaner and helps the deer cool out. Can't tell you how many bladders I've seen borken by the folks that "ream out" the a**hole. That urine can really taint the meat, so I avoid that.
If I'm boning the deer out, there are times when I don't even gut the deer. Just skin on the ground, then remove the tenderloins and bone the buck out. You can be done quicker and cleaner.
There are lots of ways to do it, and all will work. Get the meat cool and keep it clean - those are the keys.
Be sure to leave evidence of sex on the carcass if it's a buck only area. If you are cutting the horns off, be sure to leave a testicle attached to the carcass. Lots of folks get citations for that every year.
Good hunting.