I have had a lot of experience in folding paper of all different weights and materials due to my Origami habit. So, I have a lot of thoughts for folks getting started with flexagons. I love folding with expensive Japanese papers like Washi, but believe it or not, one of the best papers to use for flexagons is cheap lightweight computer printer paper. Bright white paper is nice for colors, but sometimes too heavy for the flexagons with more sides. I good inkjet printer will print nice faces for the flexagons, and if you have access to a good color laser printer, they also give nice color printouts.
Take your time, cut them out carefully and neatly. I find it best to pre-crease all the folds in the cutouts before any gluing. I crease in both directions. My favorite glue is rubber cement. It does not curl the paper, get it wet and bleed the colors. It also rubs right off the areas you do not want it. Rubber cement also gives you a few seconds to adjust the two sides of paper to perfect alignment. Overall, rubber cement really works great. For grade school kids, it may be better to use glue sticks and they also work well but make it harder to assemble due to paper curling. Glue sticks also dry very quickly and do not allow much time to adjust for mistakes in assembly. But, glue sticks are very safe and non-toxic.
Once glued, I like to again go through all the creases, back and forth carefully. Now, fold up the flexagon and glue the locking piece(s) together. I then like to go through all the sides of the flexagon and carefully adjust the creases for a perfect look and feel. I have found that well made flexagons with 6 sides will flex very smoothly and look great on all sides. Flexagons with more faces can be made, but are more of a challenge and require good assembly and folding for best operation.
One technique I use for flexagons with a lot of faces like the nonahexaflexagon is to create the flexagon with lightweight paper and create the hexagon faces in Photoshop and print them separately. I then cut out the face triangles and paste them to the flexagon. If the flexagon is made large, like 4-5 inches in diameter, the resulting flexagon will flex quite smoothly, even though it has a lot of faces. Sometimes I also create the initial template with a few millimeters of space between the triangles to allow for the hinges and thicker pats of the more complex flexagons.