I am a home exchanger, and a considering a 10 day to 14 day home exchange in Bergen, Norway. What I look for in a home exchange is a base where there are sufficient sights and activities to entertain us for an extended period of time. I know most people only take a day or a few days to see Bergen before moving on; but we enjoy immersing ourselves in the area's we visit (and taking advantage of free accomodations.) Some days we are happy to just relaxing on a beautiful beach, or visiting non-touristed villages, or just going to an outdoor market for groceries and then fixing our own home cooked meals. We do enjoy living like temporary locals. Bergen looks amazingly beautiful, but is 10 days to two weeks two much time, for even our type of travel? Also, is it economical for a family to use public transportation to get around, or should I get a home exchange that includes a car? Note: we will be a family of 4 - 6 people and the exchange would be in the summer of 2017.
I spent two nights in Bergen this past May, and I loved it. I'm sure you would too. But I share your doubts about how to fill 10-14 days there. I might have been able to fill another day in Bergen, but beyond that, I don't know what I would have done.
Though there are some swimming places, there's not any beach worth spending days relaxing on (especially compared to beaches near your home in New Jersey!). Also, you will have hordes of cruise passengers invading every day, and while there are certainly non-touristed parts of the city, I don't think these will keep you engaged for that long.
If you enjoy picnicking and barbecuing, a lot of the locals visit parks with portable and disposable grills and hang out. If you wouldn't get tired of doing that, you could fill some time that way.
There are some beautiful places you could visit on various day trips, such as fjord tours by boat, but a lot of other places well worth visiting would require at least an overnight, which I assume defeats your purpose.
It's also not a city you'd want to have a car in. Public transportation is excellent (though when you ask if it's economical, I should point out that pretty much nothing in Norway is economical; though it is probably far more economical than driving a car). If you wanted to head out of town somewhere, you could rent a car for a day at a time, but I don't know anything about what worthwhile within a day's drive round trip. Perhaps you could ask your potential home exchange people?
We too are home exchangers and get a lot of offers from people with fabulous houses for summer exchanges in Scandinavia. We have been to Bergen for a few days on a motor trip across Sweden and Norway many years ago but we haven't done an exchange there yet.
For an exchange as short as 10-14 days I wouldn't worry about running out of things to do on the west coast and fiords of Norway assuming that a car is part of the exchange.
One idea we toyed with a couple of years ago when we got a very tempting but short offer was to fly to Norway then take an inexpensive flight somewhere else for the remainder of our vacation. Bergen has a lot of inexpensive flights to the UK, Poland ($20 USD RT), CZ, etc.
You could take an inexpensive Norwegian vacation with most of your lodging paid for and then fly away to a less expensive and totally different country for a little more vacation time.