For hiking I think you'll want to have a rental car. Montenegro is a relatively small country with some very dramatic scenery, a good bit of it in national parks that do not appear to be on public-transportation routes. I was taking buses so didn't actually set foot in any of the major parks, but if you do a bit of Googling and take a look at a comprehensive guidebook, I think you'll find some useful information.
The Bay of Kotor is a must-see for anyone going to Montenegro. You may tick that box just in the normal course of moving around, but be sure you don't somehow manage to miss it. It's one of Europe's great natural sights.
For urban experiences, I enjoyed all the following, of which the first two are the most touristy:
Kotor (tourist load varies greatly depending on whether a megaship is in port)
Ulcinj (strong Turkish vibe)
Cetinje (interior town, quite different from the coastal spots)
Podgorica is the capital and a useful transport hub (you'd fly in there). It is unfortunately not a terribly attractive place, and I would definitely not want it to be my single base for the country. I spent 2 or 3 nights there to see a bit of inland Montenegro, then I moved to Kotor. Kotor is lovely if there isn't a huge ship in port. Otherwise, it's not so wonderful; the difference is really striking. I don't know what to suggest if you're hoping to spend all your time in one lodging. I'm not sure that's a viable plan. You can explore driving times on ViaMichelin.com, and Googling (bus Kotor to Budva) will probably turn up bus schedules. The buses were reliable for me, but in the Balkans it's best to go to the bus station when you arrive in town and check the schedules in person. I'm not convinced that the online information is always complete and up to date.