It is best to watch someone do this, that is how I learned (not by just watching but by doing), everybody has a different way of doing it. Just google "gutting a deer" and you will get lots of hits. This is how I do it.
Keep in mind, everybody does this differently and sometimes in different order, you do it enough you will find the way that works most efficient for you.
Turn the deer over on its back on the level ground or with the head slightly uphill.
Grab the pecker with one hand cutting it and the scrotum free with the other, down to where it emerges from the pelvis. Here you extend the knife cut to encircle the anus, cutting deeply around both tubes to free them from their channel through the pelvis. (If it is a doe, same first cut, just no pecker and a second hole to free up instead of the peter tube)
Stick your finger in the cut above where the pecker was and lift up the skin and hide and then insert the knife point, edge up, under the hide and skin, ahead of the pelvis where the ##### cut was begun and carefully split the deer to the point of the breastbone (some guys move their hand ahead of their knife to keep this raised so you don't puncture the gut, I find that my index finger on my knife hand does the same job when extended and I don't have my hand in the way (which usually means first aid kit) You can feel it where the rib cage starts and cut no further if you are mounting this. If not, I continue the cut through the ribs with the knife just off to one side so it is easier to reach in and cut the windpipe and esophagus). Be careful not to puncture anything underneath or you end up with a nice mess to clean up and a wonderful smell during the rest of the gutting (learned this the hard way)
I usually try and slice around the outside membrane (diaphragm) that attaches the innards to the ribcage so it is easier to pull the guts out. Then I reach in to the neck with one hand to grab the esophagus and with my other hand go in with the knife and carefully cut it off (the esophagus and windpipe, not my hand). Then I grab the end I just cut with both hands and start pulling towards the back, here you will encounter whatever else holds the innards in attached to the ribs and around to the back and I just slice it to free it as I keep pulling with my other hand.
Now, if you've made a proper cut around the arse end, a good pull should pull the butthole and ##### back inside the deer through the pelvis and out with the rest of the guts onto the ground. (If a pull doesn't accomplish this, a deeper cut is needed around the arse and a couple of other cuts of whatever is holding in the large intestine, etc. by the pelvis.
Flush out the body cavity by raising the deer's shoulders and letting the accumulated blood run out through the pelvic opening.
The gutting is now complete. Short and sweet (sorry not as technical as some guys will be, but it works for me)
