Skinning out your trophy
Generally, if it has antlers or horns; You'll want to cut the hide down the back, from the tail to base of the skull. A bear or other critter that may be turned into a rug has a couple of options that can be considered in the field. You can split the hide down the back, but many taxidermists may advise you you split the belly/chest from the anus to the base of the neck.
Regardless of your decision for a mount, now is the time to slow down, and remove the hide carefully. First and foremost - SHARP knives! Cut the hide in the preferred method as discussed above. If your knife cuts through the hide - stop moving the blade! Taxidermists are miracle workers with hides that have holes in them; but the best taxidermist can not UNCUT hair. When your knife goes through the skin, it will start to sever the hair above the root, and leave unnatural appearances. Again, if you puncture the hide - don't panic, just pull your knife (carefully) out.
99.9% of the time, you are not only attempting to bring a quality hide to your taxidermist, but bring meat to the freezer. With that in mind, you'll be working to preserve meat and a hide. Generally, the two do not compliment each other. Use the hide to protect the exposed meat from the elements, peel one side off, and harvest the quarters, back strap, etc then work the other side. This may help keep leaves, mud, bugs etc., off the meat.
Work you knife in ONE continuous motion through the hide, pausing, sawing or other similar movements may cut more hair than hide.
When you reach the arms/legs of the critter, you will need to envision an imaginary line down the inside of the leg and follow that line, even where the joints are... in many cases you will not appear to be making a straight line down the leg. Again, you're helping your taxidermist by making clean cuts more than anything else, they can sew and hide holes, imperfect lines -- just not cut hair.
Fleshing a trophy cape
In the field, it is important to remove as much fat, meat, viscera, etc from the hide as possible. Spend the next day, after you have carefully skinned your animal, just going over the hide (fleshing, turning, and salting). Spread the hide on the ground (cooler the better, in the shade if possible), and carefully cut off any 'steaks' or fat deposits. Try not to cut the hide, this is where many folks cut through the hide, and cut the hair roots.
Turning (lips, ears, eyelids, paws, etc.)
Not as hard as many think, just time consuming for the uninitiated. Just what it sounds like. If hair grows on the skin, that part of the skin will need to be turned inside out. Start with the ears, they are pretty big, no joints, flexible, and with all that - the 'easiest'. Begin turning the ears inside out - obviously start at the base of the ear. Use your fingers to push the ear to an inside-out position, and use a pearing knife to gently cut the flesh to allow this to happen. Take your time, you are bound to knick the hide a few times, but with practice you can do it quickly, efficiently, and without cutting the hide. Next, move on to the eyelids and lips. These aren't hard - again, no joints, smaller but still quite flexible. Take your time. If you're trophy was antlered or horned, you may be done... if it has claws - well, there are four more items to be turned. The paws/feet.
Lots of salt -- table salt is the right sized grain. Many recommend to not use iodinized salt. A 6 foot squared hide will take about 10-12 pounds of salt to evenly cover all the exposed flesh. Work the salt into the nooks and crannies of the hide.
Storage in the field
If you have a shaded, cool stretch of ground, leave your hide out - flesh side up for the day, to let the salt pull the fat and moisture from the skin. In any case, you'll need to roll the hide (skin to skin, hair to hair) and put it in a game bag - this will help keep the bugs off, keep it cool once cooled down, etc.
To Freeze or Not to Freeze