I generally take off-street bike paths when I can find them. Although I don't like places where they make street crossings every block, so I might take another low traffic side street until I can come to a section where the path goes for several uninterrupted blocks at a time.
Case in point. In Springfield, Oregon, I like the Pioneer Parkway path. However, I hate the section from Main Street, A, B, C, D, E, & F streets. Then it is good from F to Q street, or to Hayden Bridge Road.
So, for those blocks from F to Main, I'll hop over to Mill Street, taking slightly different routes depending on which way I'm going. If heading south, I'll hop onto Pioneer Parkway West for a few blocks which is a busy road, but I usually hit it between traffic pulses.
When I was in Missouri, I used to ride along the Katy Trail from St. Charles to Defiance or Marthasville, and back. Fine packed limestone trail. There was one section where there was a low traffic country road that went parallel to the Katy trail. I hopped onto the paved road once or twice because I got tired of riding on the higher rolling resistance limestone. I did have at least one driver honk at me. Obviously not someone who had just ridden 40 miles on limestone.