I worked in Bellevue for years and rode to work downtown every day. Even though I don't work downtown anymore I still ride through the city on a regular basis. There are roads in Bellevue that are fine to ride on and some I would never ride on. Motorists are used to seeing cyclists on the road and generally share nicely although you may run into the occasional jerk. I've seen cops enforcing helmet laws. You'll generally stand out if you don't wear one.

The Strava Global heatmap is a great resource to see what roads cyclists use. The City of Bellevue also has a bike map that you can download of their website.

Another good loop from downtown Bellevue is the Lake Washington Loop. Just search it online and you'll find maps, etc. plus the loop is signed.