I found a set of airbrush cleaning brushes on amazon and they are awesome. If you are using enamels you want to use lacquer thinner, but if you are using acrylics, go with an acrylic cleaner (this may be interchangeable with rubbing alcohol, but I'm not certain, the acrylic cleaner certainly smells like rubbing alcohol).
With the 350, I always make sure to clean the inside of the "needle" with a brush dipped in the appropriate cleaning fluid. I've also found that I've had paint and future dry in the little tiny hole in the head that the air comes out of. I have a cheap toothbrush I use for cleaning the brush as well and I always make sure to get one of the bristles with cleaner on it into the hole in order to clean that as well. I do the same with the cone that goes over the needle.
I use 2 jars of cleaner, a dirty jar that I dip the cleaning brushes and toothbrush into and a clean jar I use to finish up. When I think I've got it good and clean I attach the clean jar and spray onto a piece of clean paper towel or a piece of white cardstock I use to protect drying items from overspray when painting something else. If the spray is clean I'm good, if I can see color on the paper towel or cardstock I go back to cleaning.
The 350 is a great starter brush, primarily because of its cost. I have different airbrushes for different types of paint and different uses. I have 2 brushes for enamel paints, and 2 for acrylics, then I use a cheap airbrush for future. That way I'm not potentially mixing paints in my brushes. That may not be a problem if you're careful about letting things dry, but it works for me.
Groot