Before you start field dressing your deer...
Don't forget to fill out your deer tag! In New York State you are not required to tag your deer until you get it back to your vehicle, but you are required to fill your tag out immediately.
Admittedly, field dressing deer is not the most enjoyable part of hunting, but the extra time spent taking care of the meat will pay dividends at the table. Field dressing deer takes effort, so your heavy hunting coat should be removed and your sleeves rolled up so they wont be soiled.
I highly recommend rubber or plastic gloves of some kind. You can purchase game gloves found at you local sporting goods store. They are a plastic glove that goes all the up to your armpits. These allow you to put your whole arm in the deer cavity while field dressing without getting blood all over your sleeves. I take a couple rubber bands along to slide up on my arms to hold the glove sleeves up. These are well worth the buck or two it costs for the gloves.
Carefully cut a circle around the anus so it's free and can be removed from within later on. This will take a sharp knife and a little careful work. Some folks tie it off with a string to prevent its contents from spilling out and tainting the meat.
Next, remove and discard the genitals. Unless, your game laws require you to leave them intact for proof of gender.
Beginning close to pelvis, open the stomach cavity to the ribcage. After starting the cut, use the first two fingers of your other hand to help guide your knife, so that you only cut through skin and a thin layer of meat, and not the entrails.
Some hunters like to continue cutting through the ribs and skin, following the breastbone, on up to the neck. Cut the diaphragm (The the sheet of muscle that separates the stomach area from the chest cavity)loose. Sever the windpipe and esophagus as far up as you can reach.
I personally only cut the stomach up to the base of the ribcage then cut away the diaphram and reach up into the chest cavity with my knife to cut the windpipe. Once the the windpipe is free the heart and lungs can be pulled out of the chest cavity from below. You can do which ever works for you.
Tip: If you plan to have your deer mounted, don't cut it above the ribcage. Your taxidermist will appreciate this later. Leave the caping (the skinning of the head & neck) to a professional.
Allow the animal to roll on its side, and help the organs to come out. You'll have to pull a little, but they should be mostly free. Be extremely careful in removing the bladder! You must reach up into the pelvis and pinch it shut while you cut it free with the other hand. If any urine is spilled on the meat, remove it immediately with water from a thermos or a clean cloth. Clean any debris from the cavity. Any stomach contents or other substances should be removed as quickly as possible.
Turn the deer over to drain any blood that might be in the body cavity. Then it's not a bad idea to put something in the cavity to keep it spread open to help cool it down. A clean stick will do just nicely.
Separate the heart and liver if you or someone you know likes to eat them. Cloth bags are recommended for keeping these clean and allowing them to cool, but a zip-loc bag works well also.
Start dragging your deer back towards civilization after a break to catch your breath.