Salamon do a shoe called crossmax, I found these were good in mud and on road. Dependant on how you run they do them for over pronation and these are called guidance crossmax or if you are neutral they are called crossmax neutral I think. I had a pair for a year until I moved onto the speedcross for off road as I now do move fell races and cross countries.
Start slowly first off, you say you live by a track, start by warming up with walking faster for 5 minutes. Then pick something about 100 yards away and SLOWLY run to it. Walk after that for a recovery break then do the same again. I know it seems like only small amounts but you need to get your body prepared for it. Your lungs have to adjust and your body! Try 100 yards at a time and then build up so you run say 500 yards. It will come to you. I never thought I could run but now i run around 10 miles a week in a group. Build yourself up so you can run 10-15 minutes in one go; once you get there maybe join a group of joggers. I jog with 'Jog Derbyshire' but look out for groups which do around 2-5 miles in one go. They tend to have beginners/ improvers, and they tend to stop after so long for rests and wait for people. Don't be sacred or joining they will help you and remember to take it slow!
Word of advice; when running on a treadmill run with a 1% incline it helps to get the muscles ready for outside running. Oh, run up as many hills as you can find! I started in the gym, I couldn't bring myself to go outside to begin with. I started on the cross trainer then built up to running on the treadmill.