Laser cutters are awesome!
We had two laser cutters at ILM, and we used the heck out of them on styrene, acrylic and wood. Acrylic cuts better than styrene because the melting point is higher, but if you dial in your settings, styrene cuts very well. It's a balance between feed rate and power. Air assist helps a lot too.
We were constantly trying new techniques and materials on those machines. We cut rubber (sooty), leather (stinky! like burning flesh), MDF, plywood, PVC, polycarbonate (awful), glass (didn't work), etc... just to see what it'd do. Our big cutter (48"x 48") was supposed to be able to do metals as well, but we never set it up with inert gas. I never found the limit on that thing. The thickest material I ever cut on there was 1.25" thick acrylic, and it did it with out any problem. It had an indexed platten so you could load an full 4'x 8' sheet of plywood and cut big pieces. That's how all the ship bulkheads were cut for Pirates of the Caribbean.
I have a 24"x 36" cutter in my personal shop that I am upgrading the drive motors on. Like Adam Savage has said before, "once you get a laser cutter, you start to see everything as a laser cut kit".
Enjoy you new toy!