I began with craft mesh that I picked up from Walmart in the crafting section. It comes in sheets and circles. The circles are perfect enough that I don't have to do much trimming to get the size. I looked at Blanches Place for ideas on what form of hat I wanted. I settled on the 1870s hat. On the website, there are dimensions on the size of the hat. I used this as a starting step, but trimmed everything down to a shape I liked.
Then, I got out the 16 gauge wire (also can be found in the tools section at Walmart), and I attached the wire to the outside of the brim using the nylon thread I had purchased. This thread is stronger than cotton and will hold up better. I had tried on my sewing machine using a zig-zag stitch, but it didn't work well. Either it did catch the mesh or the needle kept hitting the mesh and bending. The wire is traditionally used to shape the brim when completed.
After that I sewed the top plastic circle to the brim making sure that the bottom of the side of the hat matched with the inside circle of the brim and the top circle. After this step, you have a bowl, of sorts.
Then I cut 4 pieces of fabric slightly larger than the frame: the top circle, the side of the hat, and two circles for the top and bottom of the brim piece. (I'm not very particular about the inside of the hat just yet.) I hot glued the smaller of the circles to the top of the hat. Then I glued fabric to the side of the hat making sure to role the fabric down on the top of the hat to make a nice finish.
Now, I took the two larger circles with right sides facing one another, and stitched around most of the perimeter. After flipping it inside out, I slid in the wired brim section and sewed the rest shut trying to keep the stitches as invisible as possible.
Then, I cut a circle out of the middle of the brim piece, making sure to leave enough to fold up to secure to the inside of the hat sides. I glued and sewed it altogether. After making sure all the seams are nice looking, I add any finishing touches, and there it is!
ETA: Here's another hat I recently made that didn't include the wire brim, but holds it shape perfectly.