Sounds like you should be looking for a roadster, but if you're having back problems then getting a proper fit is going to be paramount. You might want to look at a tourer... These are set up to be pretty sturdy, with a "slack" steering geometry that's stable and predictable rather than the "sports car" handling of many modern roadsters. They also tend to come with a nice, wide gearing that will get you up most anything. Cyclocross bikes are really meant for that sort of competition; they are very rugged and can be used for all sorts of riding, but also might be rather pricey.
Check out Surly.