Your best bet for framing the little 6 x 6 's and 8 x 8 's and even the 8 x 10's created on the museum boards that I am selling here is by going on line to either Dick Blick or Jerry's Art-Arama. My favorite frames are either the plein-air frames or the floater frames. Here's the links to the pages below for your easy access.
sorry no 6 x 6's in this style but the 8 x 8 and 8 x 10's are really awesome. You'll need some little nails to tack the museum board in and keep it from falling out the back side
Now you'll need a bit of a creative hand to make these work along with some velcro. The velcro goes on the back of the museum board painting and the inside of the frame itself and they stick together. Yes just like magic & presto you are ready to go - hang or can be used on a desk as they stand straight up
My favorite way of displaying these little pieces is on a miniature easel which are sold at Michael's Arts & Crafts. The Larger size is about $4.99 and the smaller is $3.99. I have only ever found these in black but I suppose they could come in other colors as well.
Michael's, or other local art supply stores, even have baby easels that fit paintings up to 9 x 12 in size. These are pretty good quality too and I use them at shows and exhibits when the pieces are on the walls. They are generally pine and unpainted. I suppose you could stain them to any color you want and or add a varnish.