You're going to have to learn to pan your shots with that camera. Otherwise a minimum of 1/5ooth for a shutter speed in order to stop the action/blur. Your camera is maxed out at 3200 as far as usable ISO goes. So practice panning!!!! Then you will probably get an acceptable number of shots at 1/200. The background will of course be blurred but your subject should be sharp. You also will have to use spot focus or at best the 5 point auto focus settings.
It actually goes to 6400 , would it be best to use that over 3200?
You can try 6400 but you will end up with more noise, what it would get you is the faster shutter speeds you need. At the skill level you are working at I wouldn't waste too many frames trying to pan the gymnasts at 200th sec. You may get lucky on a couple, but for the most part you'll end up with a lot of deleted images.
DUH! That is why you practice! What I described is easily learnable in a couple or three hours. It is a technique even the top sports. action photographers use. Maybe you do not want to spend the effort, but why advise someone else that it won't work?
You're going to have to learn to pan your shots with that camera. Otherwise a minimum of 1/5ooth for a shutter speed in order to stop the action/blur. Your camera is maxed out at 3200 as far as usable ISO goes. So practice panning!!!! Then you will probably get an acceptable number of shots at 1/200. The background will of course be blurred but your subject should be sharp. You also will have to use spot focus or at best the 5 point auto focus settings.
It actually goes to 6400 , would it be best to use that over 3200?
On that camera 6400 is not a usable ISO, 3200 is barely so. Re-read what I just wrote in my previous comment. Do that and then come backk with some examples...there is NO easy lazy solution for you other than spend a LARGE amount of cash for a new body. Why do that until you master the techniques used by ALL the good sports/action photographers? Just buckle down and do the work. You have the equipment, you just don't have the skills yet.
There IS one more thing that will help greatly. Get a good flash (I recommend the Yongnuo 600 @ around $100 it is the equal of the Canon 600EX). If necessary to shoot beyond 40-50 ft put on a Better Beamer flash extend (~$40) dead simple and it will stop the motion easily at 1/200 or even faster using its hi-speed synch.
DO not be concerned that it is too complicated. JUST ASK. There a dozen or more photographers on here that can explain it simply and easily.