I'd go cyclo-cross bike. It would be lighter on average than a mountain bike. Which is one of your concerns right? With back probs. and flights of stairs to climb. Lots of manufacturers make them - one thing - if you are short, most cyclocross bikes don't come in a small enough size. I think Surly's CrossCheck is the exception. If you are not, you don't have to worry.

Trek makes the XO-1 and XO-2 www.trekbikes.com, Specialized makes the TriCross www.specializedbikes.com, Jamis makes some fine bikes - Nova Race, Nova Pro, and SuperNova www.jamisbikes.com, and the other suggestions are great too!

Do your research. That is the best advice I can give you. Go to your Local Bike Shop (LBS for short on the boards) and ask questions. Do not go to a dept. store - steer clear of cheap. Quality does matter in bicycles. Or, if your budget is big go custom. You have lots of options. Happy bike-shopping to you.