What type of stone should I cut first?
Quartz is often the stone people recommend as a good place to start, when asked. There are several reasons for this, it is inexpensive, easy to acquire (at least in clear material) and there are a lot of common designs that are set up for it available.
As far as I am concerned, the first type of material a new cutter should cut is Beryl or Quartz.
Beryl and Quartz are not expensive in the very pale colors ($0.20/gram to $3/gram) and fairly easy to find. Either material will work well.
Note: Tourmaline and Garnet are also fine for a first stone, but they are usually more money and that is why they are generally not recommneded for a new cutter.
Both of these materials will be worth some thing when you are done cutting. To me it is important to cut some thing worth while because of the many hours of work it will take you to facet a stone.
I prefer that people I teach start on a natural stone, but a laser glass is a good choice if they really want to cut synthetics. Stay away from CZ, Corundum (mane-made Sapphire), or Spinel, these are much harder to work with and are can be hard to polish, especially for someone new to the hobby. They also require more laps and time to cut and polish.
I recommend that you start with a moderate size stone 5-20cts (1-4grams), this gives you room to make mistakes (you probably will) and correct them.
For your second stone, if you want to try something new I would say Garnet is a good choice. After cutting Beryl, or Quartz, a Garnet of some type or maybe a piece of Tourmaline is a good choice to cut next.
Topaz is cheap in the light colors and blues, it is a good choice after you have cut several stones and are feeling a little more confident. Topaz is a little bit harder to work with because of it’s perfect cleavage, and it takes a little more effort to polish. So I would say cut at least one stone each of Quartz, Beryl, Garnets, and Tourmaline, before trying Topaz.