Start by cutting the penis and testicles to the side. You need to grab the penis and testicles and lift them away from the body before you make the cuts. You don't want to cut the gut on accident and You don't want to just whack em off, or tug to much. Stop that snickering' in most states you need proof of sex. . Check your game laws in your state to see what is required for proof of sex. If horns are good enough then these man parts can be cut off. 

Now the anus needs to be cut out as well. What you do is carve around the anus " deep" The anus is attached through the pelvis of the deer. So we carve it out like a pumpkin pie. Just circle it with your knife. Just cut enough to free it away, you'll be able to tell. Now we have a carved ring around the anus. "stop laughing'"

Now if you don't plan on mounting your deer then we can do it the easy way. What we do is slide our fingers in pointed towards the chest, in a peace sign fashion, and lift the skin away from the stomach. 

While you are lifting you insert your knife between your two fingers and glide it up nice and easy until you get to the chest bone. You don't want to cut the gut because it will smell awful and get germs and gunk on the meat. So be careful. That's why our fingers are in there, to pull the skin away from the gut, and to push the gut away from the skin. Now take your fingers out and cut right along where the ribs come together on the breast bone,  and then you got something that looks like the picture below..

Now folks, this here is a female deer so don't be writing me any e-mails telling me the obvious.  Of course on this deer we didn't have to slice around the penis and testicles. " stop the snickering'" What you do is carve out the vagina like we did the anus. " I mean it quit laughing'"  Now slice down the middle of the pelvis not penis so it splits open " look above" 
Now once we have the deer or other animal split open like this we need to get up into the throat and slice away the windpipe and esophagus. Just reach in there and do it.

Now some of the organs will have to be cut away to free them. Just kind of tip it to the side and look and see what is holding the stuff in, and slice it away as careful as possible.  But don't cut the loins. The loins are good eatin'. and then free the intestines and bladder. The anus will just be pulled through the carving we made around it. Be careful not to get any pee or poo on your meat. A garbage bag twist tie comes in handy for closing off tubes. Tip your game up so the blood will drain and wham bam you got yourself a gut pile!
 

Here these young men are spraying out the inside with a water hose. You want your meat to be clean. It is a lot easier to hang the deer or other animal from a hang or a tree and wash it out, but the deer hang here had gotten some poisonous chemicals on it so they stayed away from it.

If you plan on mounting your game you need to read this by Larry Reese

As always you need to know the anatomy of the game to ethically harvest it and gut it. take a look here. National Bowhunter Education Foundation.