I gut in the field, get it home, hose it out good. If it's warm, I hang it, skin it, quarter it and get it in a fridge asap.
If it's cold, which it usually is since I don't want to fill my tag early on, I gut it in the field, take it home, hose it out, and hang it. Leave it till the next day or two when I have time , weather permitting.
As for my butchering process, I use to hang by the neck or antlers if it was a smaller buck, quarter it, and cut it up in the kitchen.
The last 4 years or so I've hung them with a gambrel, and pull the cuts off the carcass, sort of like filleting. Get alot cleaner looking cuts, less tendons, less waste, and saves a lot of cleanup in the house. Picked this method up from a video on youtube.