You might consider a used mountain bike--they're very widely available here in thrift shops (Salvation Army, Good Will etc) for $25-$75 and sometimes cheaper. I bought my wife a near-mint Specialized Hard Rock as a backup to keep at her office for just $10 last year. It didn't need anything but air in the tires and lube on the chain. If you spend, say, $50, you could put on road tires (from any bike shop) for another $20-$50 and possibly fenders ($20-$30, and well worth having in winter), and you'd be rolling for $100-$150.
It helps to know what you're looking for, but many thrift shop bikes are in great shape, donated by people who bought them 10 years ago, rarely used them and finally got tired of walking around them in the garage. Specialized, Diamond Back, Trek and Bridgestone are some reliable brands. Be wary of Huffy, Murray and Magma, which are junk. If you don't see what you want on the first trip, wait a week and look again. Turnover, at least in my area, is pretty rapid.
Before you buy, Google something like "bicycle fit" or "fitting a bike" to get a rough idea of what size you need. Fit's not critically important for the kind of short-range cycling you're talking about, but it's always best to be comfortable, and if you get enthusiastic and want to stretch it out, you'll be glad you got the right size.