I typically make my jigsaw style puzzles out of 1/4 " stock plywood is best poplar if you can find it is the best.
I use mod podge to adhere my picture to the wood others use aileens tacky glue. Then cover with a product caller triple thick glaze. Let dry for 24 hours and cut away. I suggest starting out freehand and never going back to grids. There are no mistakes because each one is an original. Cutting speed is up to you and your level of expertise. Blade size I use a superior puzzle blade for adults and up to a #5 or even a #7 for childrens puzzles.
I prefer flying dutchmen ultra reverse but there are many providers of quality blades out there so pick some and have fun.
I give my puzzles to the recipient unassembled with no pictures. You can use many techiniques, color line, figurals, drop outs and size of pieces can vary as well. Most important is too have fun.
if you are printing a picture with your home ink jet use at least #24 paper I prefer #65 or 67 don't remember right now so I do not have the dreaded bubbling of the print. You can also use high quality photo paper for printing...I use the regular paper because I'm cheap LOL & I do several coloring pages and use crayons to color them first for tray style puzzles as well.
hoe this has been helpful. There are lots of puzzlers on several other forums as well with great ideas. Look around you will find all kinds of help!
Dale here can help with stand up style puzzles as well...