a mountain bike would probably be a lousy idea for your needs. it might put you in the riding position you want, but the heavy-duty frame and fat, knobby tires will just needlessly slow you down. conversely, an inefficient bike means you burn more calories! look for a pavement-oriented bike with smooth tires (not big knobbies for dirt and mud.)
it sounds to me that finding a bike that is comfortable for you after your surgery, and you can only figure out what that is by riding a few bikes. in extreme cases, someone with back pain would need a recumbent bike, less severe cases would want a "comfort hybrid," which is characterized by tall handlebars and a big saddle to allow a very upright position while riding. next would be a "fitness hybrid" or "flat-bar road bike," which, when fitted correctly, is aggressive but with a somewhat upright position. if you want to ride long distances, a touring bike or a road bike with "endurance geometry" might suit you. this is a bike with drop bars that are set up tall so relax the angle of your back. just stay away from "race-oriented" road bikes. most of us have no use for a racing bike, especially someone with back issues.
You can also take just about any bicycle and put a taller stem and/or taller handlebars on it. if the bike fits and you really like it, ask if a mechanic can re-fit the bike with a tallers rise/ shorter reach stem and taller bars if it has a flat handlebar. the bike shop might charge you a little extra for the parts and labor to do this, but they should also give you credit for the parts they have to take off the bike.
in the end, you're going to have to visit a few bike shops and test ride a few bikes. I recommend a few different bike shops because, having worked at bike shops over the past few years, I find that the level of competence among bike shop salespeople varies widely. go in and explain your situation and ask who the most experienced bike fitter in the store is. any bike shop worth anything will have some sort of basic bike fit knowledge and will let you test ride a few bikes.
how many bike shops do you have in your area? what bike manufacturers do they sell?
what is your budget? this should be the last question any bike salesperson asks after they figure out what style of bike you need, but be aware that something decent and durable will cost $500-1000. include a helmet, water bottle cages, blinky lights, and some cycling shorts (yes, you want the padded shorts!) in your budget.