I use Flying Dutchman reverse SkipTooth blades. Like Jimbo, I use #7 for the outside & # 5 for the internal cuts. I mostly use 3/4" pine shelving for my puzzles. I get seconds that are cheaper and arrange the pattern pieces around the knots/imperfections. Make sure you check the reverse side of the board for problems. I have cut a puzzle out only to find it had a "nice" strip of core wood right across the back!
I sand around all the cut pieces lightly, to get rid of any fuzzies & to slightly round the edges, either with the dremel or by hand with a small roll of 240grit sandpaper.