If you're only training for half of a mile to one mile, running shoes will make a difference but they won't be as critical as if you were logging a lot of miles. Still, I'd recommend you go to a local running store where they watch you walk to determine what type of running shoe will work for you. They'll cost $80-120
Buy yourself a stop watch. Find a track or a trail with distance markers. Divide your "competition" distance by four:
.5mi(~800meters) will be 200m
1mi will be 400m
Warm up jog for about eight minutes. Then run each of these splits 20% faster than your goal time. For example, if your goal time for a mile is 6min, run each 400m/.25mi at 72 seconds instead of your "race time" goal of 90 seconds. Obviously start at a pace better suited to where you are right now, and then adjust as necessary.
This is a good sample workout