Also, talk to your instructor about how he/she would feel about you taking some pictures with flash. I study Kuk Sool Won, and I often took pictures of training and testing. For testing, I'd never use flash, as the students had enough to worry about without a bunch of flashes going off in their faces. For practice, I took most of my shots without flash, but also some select shots with flash as I felt appropriate.
But just as Chet said, the lighting in most gyms is much worse than it usually appears. We had some pretty decent lighting in the gym where I trained, plus some windows that let in natural light, but I still ended up shooting at ISO 1600 and f/1.4-2.0. Speaking of which, if you're consider doing this more in the future, a 50mm prime lens is great for indoor martial arts, and not very expensive.
For shutter speed, it depended on the shot, but I actually liked a little motion blur in many instances. We do a good amount of throws and take-downs, and I used 1/125-1/250 quite a bit. . .probably closer to 1/250 for throws, as it will keep the person throwing pretty clear (especially the lower body), while the person being thrown has some good motion blur, which adds some impact to the photo. If you really want to freeze the motion, 1/400 - 1/500 should work fine for throws, especially at the beginning of the throw before the person really starts picking up momentum. I seldom went higher than that and just accepted a tiny bit of blur with really fast motion, but to freeze a really fast punch, kick or even throw, you may certainly need a slightly higher shutter speed, which is probably going to necessitate bumping the ISO up to 1/3200. . .which in the newer cameras is certainly feasible, especially with some good noise reduction software.
With flash, you can of course go with a much slower shutter speed, depending on your ISO setting and f-stop. A low ISO/high f-stop won't let in much ambient light and will freeze the action very well. A higher ISO/lower f-stop will allow more ambient light in and add some ghosting from the motion, which can actually be desireable to once again show the motion.
That's probably more than you really wanted to know. . .