Within Rovaniemi itself, the bus network seems to be quite good considering its (small — except by Finnish standards) size. The route to the Arctic Circle where Santa Claus Village is (Napapiiri) is operated roughly hourly on weekdays. Departures are a lot less frequent on weekends though, and buses stop operating in the evening (something like 6 p.m. as per random samples). Other lines (again as per random sampling) are operated a lot less frequently.
To plan your routes, you can use the local version of matkahuolto which is even available in English. Simply type in destination and location and it will even tell you how far you have to walk.
Also, Rovaniemi has an English-language page on public transport which will give you a few important routes and destinations. Note line 8 going to the Santa park. Clicking on any line number’s link will point you to matkahuolto again to plan your route.
Urban Rovaniemi is not huge so if you want, you can reach anywhere with a good walk. But that is not an option for everybody. So as soon as you want to get somewhere comfortably after hours (and I see northern lights in your list) you may well want to rent a car or rely on taxis. Same is valid as soon as you leave Rovaniemi for any other place in Lapland as public transport out of town is really scarce.
(Of course, Helsinki-Vantaa to the central railway station (rautatientori) is easily done at any time of the day by direct bus or train connection.)