The UNESCO listed harbour fronted, colourful, wooden buildings of Bryggen is a good place to start exploring Bergen. There are also some great museums and mountains to climb [well a cable car or funicular] providing stunning views of the city. Bergen is also the starting point for wonderful The Norway in a Nutshell tour or Hurtigruten Coastal Voyage.
We stayed at the Comfort Hotel Holberg [previously a Thon Hotel] on the edge of Bryggen which, although rather ugly on the outside, was comfortable, conviently located and was well priced [for Norway]. Airbnb may be a good option also if you need to keep the price down.
We grabbed the Flybussen from the airport into Bergen which worked well for us, the bus stop was only a minute walk from the hotel. There are plenty of bars and restaurants of course, click here for my suggestions.