One of the first projects I made with my laser cutter was a set of cube fridge magnets. The process was fairly simple but I went through a number of steps to get the look I wanted.
Paint the wood first
It may seem counterintuitive, but I actually painted the wood before laser cutting it. The trick is to use a water-based non-toxic ink. Since I'm screen printer, I used Speedball acrylic screen printing ink for wood and paper. The colors are great and it's inexpensive. Make sure you get the ink for paper and not fabric.
WARNING: Never use oil-based or solvent-based paints, stains, or varnishes before laser cutting. They're flammable and could release toxic fumes. Always check the SDS sheet before using.
Avoid burn marks
The last thing I wanted to do is mar my beautiful paint job with laser burn marks so I covered the wood with some transfer tape that I had for transferring vinyl cutouts. Transfer tape works great for protecting wood and acrylic when laser cutting but doesn't stick very well to wood so I recommend a high-tack tape and applying it with a squeegee
If you do get a few burn marks, you can often wash them off with warm water and a sponge or a little Gojo pumice soap.
Cut the pieces separately
I find that in order to get a really clean line between colors, I need to cut out the colored pieces separately and then glue them together. Trying to paint separate colors onto tiny pieces of wood is frustrating and tiring (at least for me).
Cut the backs for the magnets
In order to get the magnet to lie flush with the wood, I cut another piece of wood for the backs of the magnets. I accounted for the kerf so I got a snug fit for the magnet.
Glue everything together
Once everything was painted and cut, it was time to glue everything together. I found that super glue actually works great. The bond is strong enough for small pieces and it dries super fast. I tried wood glue first but it dried too slowly.
Final varnish coat
After everything was glued together, I applied a final coat of varnish. After trying a bunch of varnishes, I settled on General Finishes High Performance Top Coat in satin. It's non-toxic, low odor, and goes on smooth. The subtle shine really brings out the colors and makes everything look finished and protects the colored paint underneath. The can recommends a 3-coat minimum but I get good results with just one to two coats.