I agree, gutting should be done as just as soon as you find the animal.
First, position the deer on its back. If it is a buck with a reasonable rack, I like to hook it's front legs under the main beams (like putting your hands behind your head), this will help keep the deer from rolling on its side.
It is nice to have a length of small rope or chord with you to tie front, and or back legs open as well.
Some folks cut off the tarsal glands from a buck. I have never done this myself. I just try not to touch them.
Start at the brisket, the point where the rib cage ends. And using the point of your knife, gently pierce the hide (make sure you keep the blade of you knife facing "up") this will help avoid sticking the paunch (stomach). By lightly tapping the handle you can get the knife started. After you have a small incision here. You can use a gut hook (if your knife has one) and slide the blade down the center of the belly. If you don't have a gut hook. Again, (keep your blade facing up) use two separated fingers to guide the blade down through the hide. Keep upward pressure on your fingers ,to keep the Blade away form the paunch and intestines.
Make the incision all the way to the hips of the deer. Disconnect the tissue around the male parts and pull them down to this point also.
Then go back up to the deer's chest again and pierce the diaphragm. (the membrane that separates the vital organs from the stomach and intestines). If you made a good killing shot on the animal, this is where all of the blood will have pooled. Make a complete circular incision, all the way around the diaphragm. Then reach way up into the chest (be careful with your knife), and holding it with one hand and knife in the other, sever the windpipe.
You can carry a plastic cable tie to tie off the windpipe if you like, to prevent any stomach contents from tainting the meat.
You can then pull hard (cutting where you have to) and pull the vital organs and diaphragm out of the deer.
Next go to the rear end of the deer. You next have to make an incision around the anus of the deer. Basically coring it out like an apple. There is actually a tool for this out now, but I have not used it myself. Again cable ties or small chord rope etc can be used to tie off urinary and bowel tracks , to prevent spillage and contamination of meat.
You can tie off the bladder inside of the animal, or pull it out as far as you can and then tie it off. Either way you will have to pull everything "back through" the opening and remove it with the intestines and stomach. Again always be careful with you blade around the intestines and stomach.
Note that you will have to do some extra cutting in behind all these organs to remove the tissue that attaches them to the back of the deer. Cut as you pull.
If you have access to clean water wash the cavity out. I wouldn't use water with an unknown history, such as just a brook etc. A rag or t shirt to soak up blood in the cavity is good. I also try to get as little dirt and leaves etc in the cavity while dragging as well. Better, "not" to drag if you can…although most times you must.
You can wash the chest cavity out a bit when you get home, But I try not to use too much water. Its better to let a natural hard membrane form on the exposed meat. One thing that I recommend especially in bow season is "ice". Buy a couple of bags and stuff them into the cavity. Leave them there overnight if you cannot skin the deer immediately. In warm bow season I don't like to leave the hide on any longer than necessary either.
Hope this helps...and you get a chance to try it!