Here's a look at some of your criteria:
Affordable house
Here are the median house prices for each of these areas (including unincorporated areas in the same ZIP codes):
Bremerton: $245K
Port Orchard: $266K
Silverdale: $332K
As you can tell, Silverdale tends to be a more affluent area. The outlaying areas on the hill are full of more wealthy folks and military. Port Orchard is more affordable, especially the outlaying areas, which contain a lot of houses in the low $200K range (around Parkwood). Bremerton is least expensive, and the city itself is full of starter homes for working-class military and port folks.
Safety
Here is city-data's index for city safety. Higher is worse; national average is 325.2.
Bremerton: 646.2
Port Orchard: 526.6
Silverdale is not available, but I believe it is a bit safer, which goes along with it being a bit wealthier. I think it is around national average at highest.
Good School
GreatSchools gives the local high schools the following 1-10 ratings:
Bremerton High School (Bremerton): 5
Central Kitsap High School (Silverdale): 9
Olympic High School (North Bremerton/South Silverdale): 8
South Kitsap High School (Port Orchard): 7
With the schools, you'll find a lot of correlation with income. Silverdale does best, followed by the rural area between Bremerton and Silverdale, then Port Orchard, then Bremerton itself. I've heard anecdotally that the Bremerton School District is "troubled."
Employment Opportunities
I can't attest to this, but all of these areas are within reasonable commuting distance of each other. The job markets probably are too intertwined to get meaningful data from each. Harrison Medical Center has branches in all three cities.
Hope that helps you out.
