I'm not sure how good Bremerton Transit is. I would guess it mainly runs to the shipyard. That's probably another difference between NYC and Kitsap county--almost everyone has a car in Kitsap.
Public transportation is not nearly as developed here as NYC. You have to be very conscious of where the limited number of bus lines are when selecting a location if that is how you will be getting around. Have a look at http://www.kitsaptransit.org/ for bus routes. If you are here because of work, you will need to see if public transportation can even get you there.
If you need help finding rentals, Bobbi Neal would be a good resource. email@example.com
Good luck. Kitsap County is a lovely area and close to Seattle.But it is much more like rural upstate New York than the city and the surrounding suburbs.
I haven't lived there for over 30 years, but I doubt you could find an apartment house that is over 4 stories high, unless maybe it's something overlooking the water, in which case it might be slightly higher. The population density is low.
Here are some resources for you to get to know Kitsap County and see what you think are better neighborhoods for you and your needs.
You can get a feel for different parts of the county and where you feel might be your best fit.
Do you want to live in a "small town" with close access to restaurants, libraries, schools, groceries?
Do you want to live on some "land" with a couple acres around you?
Do you want to live near water? Perhaps near a ferry where you can jump on a ferry and commute to Seattle or just go enjoy the day?
"Best neighborhood" means something different to everyone.