Welcome to Kitsap County! I moved to this area from Hawaii too. I was a little reluctant at first to get into the water, but once I did, I decided this was one of the top diving spots in the continental U.S.
You won't have a lot to choose from in the immediate Bremerton area. There is Sound Dive Center and Sound Dive Center. I bought a DUI drysuit (and a couple of steel 80's and many other things) from them over the years and have no regrets. It's a fairly big store because it serves the region as well as Navy and commercial divers (geoduck harvesters, etc.)
There is also a dive shop on Bainbridge Island. It's smaller, cozier, and I now do most of my business there because it's closer for me. But it's a bit of a drive from Bremerton. There used to be a shop south of Bremerton, in Gig Harbor I think. I've never been there and don't know if it's still there or not. There's also a small store in Port Townsend, but that's an hour or so from Bremerton.
Across the water in the Seattle area there are 2 divestore chains as well as lots of independent stores that come and go. They all have strong and weak points, but I can't think of any that I'd advise someone to totally avoid.
Good luck, and if you have any other questions about this area, feel free to ask. There's also a Pacific Northwest forum at:
http://www.northwestdiver.com