$500 is a tiny budget for a new bike but a huge budget on Craigslist... You can ride nearly any bike for 3 miles, it will take ~15 minutes.
Remember to budget / plan for...
a lock and a place to lock it
a way to see and be seen at night. The days are getting shorter!
a way to carry your stuff - plenty of options
a way to stay warm and dry. Could be layers and fenders, could be a bus pass
a way to fix flats and maybe worse - at 3 miles I might walk it!
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Genesis 49:16-17
"Well, well!" said Holmes, impatiently. "A good cyclist does not need a high road. The moor is intersected with paths and the moon is at the full."