So you want to make your own custom puzzle but not sure where to start? It depends on what you are trying to accomplish. For starters though you do need a few things.
1. Puzzles to spare - What good does desire to make custom puzzles do if you don't have the spare puzzles to mangle? I keep most of my puzzle parts and spares in a large box. Eventually I'll use a more elegant container as a beat up old cardboard box is about as dirtbag as you can get. At least it's a step up from the plastic grocery bags I was using for a short while. Did I just admit that?
2. A handheld multipurpose tool - I happen to have a Dremel but there are any number of brands that do similar work. This is a really useful tool when it comes to creating puzzles as long as you have a variety of bits to work with although not all modifications rely heavily on this. The bits shown are the ones I use 95% of the Time. From left to right: Coarse drum sander, flat cutting wheel, wood/plastic shaping wheel, course routing/engraving bit, large cutting drum, small cutting drum, and small routing/engraving bit.
3. More hand tools - I've always had an affinity for sharp objects so naturally I use them during builds. My Swiss army knife is quite capable of the light use during puzzle making. Similarly the folding knife is used in situations where more precise guiding (and an easier to grip tool) is required. So why not X-Acto blades or similar? Beats me. The flashlight also allows me a closer look into dark crevices. Yes, that is necessary occasionally.
4. Glues and epoxies - I use two part quick setting general purpose epoxy for most cementing chores. It takes about 5 minutes for the epoxy to start hardening to the point where it no longer pours easily. After about half an hour I usually trim or scrape off any excess, which easily comes off of puzzle pieces. Avoid the whitish colored plastic specific epoxy. It may or may not work better, but it does give off a really foul, toxic odor which is probably not the best for you. The epoxy pictured here (two different brands, same general stuff) is very mild smelling and easy to work with.
I also employ epoxy putties like Milliput, Magic Sculp, and Apoxie Sculpt. My experiences working with those materials can be summed up here.