The best way for you to find out what you like is with experimentation. What we have here is a few tips about hanging that will help with the process and therefore your experience with game.
It is important to keep the hanging birds as cool as possible - therefore as soon as they have been shot keep them separated to help with the cooling process.
When hanging make sure the game is covered or somewhere the flies and other insects cannot get to. Coolers and cold rooms protect the birds and help with the process of cooling.
Make sure you know where the game has been shot – if it is a gut shot or badly mangled then do not hang it, instead use it for a casserole or stew and chop it up immediately. Gut shots will spread the intestines and stomach all over the meat in turn spoiling it, hence wanting to get it out as quickly as possible and cleaning it.
Game to Eat suggests game birds should be left for 5-7 days at around 5oC (again personal preference). If warmer than 5 oC, leave it hanging for less time and vice versa. Pigeon on the other hand can and should be eaten straight away.
Similar rules apply for the hanging of ground game with the addition that they should be gutted straight away before hanging. When gutting a deer it should be tested for TB and other diseases, further details on this can be found on the British Deer Society's website.
The bigger the animal the longer you should leave it to hang. Game to Eat suggests up to 14 days for a red deer at around 7 oC going down to 3 days for a muntjac. Once again if it is warmer bring those days down, however the longer you leave the more tender the meat will become.