Another vote in with the others. At the very least, buy a used bike from a bike shop. At least they will have gone over it and adjusted everything properly.

For a one mile commute, any bike will be fast enough. You won't even have a chance to work up a sweat.

It sounds like you're kind of out in the sticks, so a road bike is probably a good choice. You will need to put aside $30-50 for a good lock (U-lock + cable) and a helmet. ($40 min.)