Norway without car and with a tent?

Firstly buy a guide book, secondly why have you chosen to go to a country you know nothing about?

Is it expensive yes but a bit of planning can mean getting fairly reasonable transportation, free camping is common as over most the Nordic countries, cannot remember the exact rule as long as you are out of sight of a house you can camp.

There is no such thing as a must see must do because one persons must do/see is not the same as anothers.

There are lots of organised campsites, and they aren't unreasonably expensive, but in the case of larger towns especially they can be more than walking distance from the town centre. Free camping, while legal, can be difficult in practice. Flat pieces of land are in short supply in Norway, and thus many of them are already occupied by someone's house or field, or are rocky, marshy or occupied by tall vegetation. Sometimes I've had problem enough finding somewhere to free camp when I've been on a bicycle, let alone without wheels.

A combination of hostelling, camping in organised campsites and wild camping will keep your budget reasonable, but not cheap.
Transportation is expensive too. To find a nice camping place, be ready to walk or even hitch a lot.

Norway bus Express (http://www.nor-way.no/) takes you almost anywhere in Norway,and you can make your own stops. You can stop in villages,cities and in the montains. Out in forests or up in the montains you can camp free as long as you are more than 150 meter from nearest house, and not more than to nights the same place.
Close to many villages and in the cities you will find places for organized camping for between NOK100 and 150 per night for tent.