If your deer is gutted properly during your field dressing process, you shouldn't have any "gamey" taste. We've never needed to "brine out" our venison...heck, would you soak all the blood out of a good beef steak? No! That's the best part and where all the flavor is from!
I agree with the post about skinning with the golf ball method...by far the fastest and less hair residue, and can be utilized just as quickly if the carcass is warm or cold. We use a hammer....no golf balls out in redneck country.
With a doe, we remove the milk bag first, circumscribe the anus and then proceed to gut. If the hocks are moist we use a separate knife to take these off first(don't want these to come in contact with anything, trust me!). In either sex, a separate knife to cut off the penis, or milk bag, and cut around the anus, and another knife to complete the gutting process. We have found that a knife tainted with exposure to the secretions in the hocks or the other areas listed can give your meat that "gamey" flavor.
From kill to quartered and in the fridge, chillin', takes us around 45 min. Thats with knives only, saws are only for taking the legs off!
PC, for a city slicker you did okay! You follow the right tips and you could become a pro! Oh...and we gut ours in the field, not hanging. We've found, the sooner you can get these out, the better the taste of your meat. Think about all these deer strapped to a warm hood of a truck all day, improperly field dressed, skinned eventually and then someone complaining that venison has a gamey flavor. Is it any wonder?
BTW, in the absence of a grinder, canned deer meat is a great option!