The reason I suggested the 85 f/1.8 and the 135 f/2 is that they have an f-stop advantage over even the 70-200 f/2.8, and that is a factor of two in shutter speed, which can be crucial. Having said that, I have shot some BB pictures at stop points using the 70-200 f/4 at 1/200 at ISO 3200 on a monopod ( I needed 170mm to get enough reach, and the lighting was better than normal). The main disadvantage of the 70 - 200 f/2.8 ( with all due respect to those devoted to that lens ) is both its weight and its size. 75 per cent of my competition gymnastics pictures are shot with the 85; the 135L is nice to have when you either want a more intimate framing or you need the reach. A 40D with the 85 f/1.8 mounted is relatively light weight and unobtrusive. All four of these lenses ( 85 f/1.8, 135 f/2, 70 - 200 f/2.8 and 70 - 200 f/4) have very fast, accurate AF in my experience.
As I pointed out in my original post, and others have reiterated, you need to shoot in RAW mode, not JPEG. This to preserve the maximum dynamic range for post-processing using DPP or whatever other RAW converter of your choice. Some of this is for exposure adjustment, although usually not that much ( I usually end up pushing it down 1/3 stop or so) but mainly for white balance adjustment. All gyms are different in the type and quality of the lighting ( some have skylights which mixes sunlight in with the rest of it ) and I find myself adjusting the WB on a shot to shot basis in many gyms ( usually you can find some nominally white or gray object either in the scene or on the gymnast to use as a test card). You also need to pay some attention to the metering on the different events; I know of at least one gym where the exposure changes by 1 1/2 f-stops from one side of the gym to the other.
Take lots of pictures; there will be some keepers in there. Pro photographers shoot hundreds to thousands to get the really good ones.