A rigid mountain bike can make a good commuter with a few minor modifications. Buying used would allow you to get more for your money. They are sturdy enough for rough roads have good brakes and gearing, a comfortable riding position that is practical for riding in traffic, and usually have fittings to allow installation of fenders, a rack etc. You should replace knobbly tyres with fat slicks for road riding- they will make it easier to pedal and give you the best on-road traction, wet or dry. I'd consider full-coverage fenders to be essential for keeping road water off you (water falling up seems to get you wetter than water falling down). Obviously, if you want to take it off road all you need to do is switch back to treaded tyres.

Alternatively, an English roadster might be a good option. It's essentially what I've converted my ex-mtb into, and gives you a sturdy machine that is ideally suited to city riding.

Remember to budget for a high-quality lock to deter thieves, and a good lighting system front and rear. Wear any safety gear you feel appropriate for the riding you do. I wear gloves on the road.