Any entry-level hybrid bike from a respected brand would do the job. $300 will buy you a capable and reliable bike. As you pay more you get a lighter bike and lonnger lasting components. Get it from a proper bike shop where they can advise you on the correct size and fit.
You can get a women's style open frame if you want. At 5'8" you wont have a problem getting the right size.
A utility bike needs medium width tyres (32-38m), a rear luggage rack and fenders. You also need to budget for lights, helmet, gloves and lugagge bags.

If you want to commute distances over 10 miles then a sportier style of bike may be more appropriate.
Hub gear bikes are useful for the low maintenance and make good city bikes. The cheaper ones are not so good on longer rides and you have a limitted gear range.

If you get into cycling as a sport you may want another bike but a good, practical utiltity bike is always useful to have.