I think you would be best served by heading to a local bike shop. You can get any range of opinions of what you should get but in the end what you like will be most important. Also realize that what you like may change after you have been riding for a year or two.
Go to a few bike shops, let them know what kind of riding you want to do. They'll put you on a few bikes to test. Don't buy that day. Go to another bike shop or two and try their offerings. Go through this cycle once maybe twice to help solidify what you think you want. Try those super expensive bikes that you know you will not buy because there maybe something about them that you'll figure out you would like to have. After you figure out what you would like, then start factoring in price. You'll probably end up with 2-4 bikes that you will be deciding between.
With the hills, I am pretty sure you will want gears. Whether it is an internal geared hub or cassette/freewheel you can decide later.
With weak knees, make sure you are spinning on your bike. Don't push big gears, that will aggravate your knees. Gears will let you choose higher cadences to a point, depending on how hilly your area is.
Tires can be fairly easily changed: fat, narrow, medium; smooth, tread, knobbies; hi-pressure, lo-pressure, etc. Work that in after you figure out what bike you want.
Bikes are pretty close in efficiency. The biggest factor is aerodynamics at speed. However, friction from a poorly assembled bike does not help. A super expensive bike can be poorly assembled and be less efficient than a well assembled cheap bike.
Good Luck