Getting around in Norway by public transport

I am planning to go to Norway for two or three weeks next year.

Is it possible to get around the country by public transportation (busses and trains) only? Are smaller towns connected by public transport or would one have to rely on other means of travel to visit small towns, especially in the northern parts of the country?

I am thinking of bus connections like Greyhound in the USA or Stagecoach in the UK. What about train connections? Is there a public railway system and how developed is it? (Thinking about Iceland here, which doesn't have a public railway system)