The AF will be, whether f4 is good enough or not completely depends on the venue lighting and the type of sport.
- In a majority of venues, light is either low, poor quality/partial spectrum, or covers areas spottily.
- Some action is faster than others, requiring faster shutter speeds.
These two factors will drive your ISO and shutter speeds, and you may find that f4 won't allow you to reach your goal of either proper exposure or stopping motion.
I shoot in a very well lit venue compared to others (professional court, televised sometimes, etc), and I rarely shoot at f4, I am always at f2.8 or f3.2. However I am running 1/2000 for a shutter. If I ran the shutter at 1/1250th (the lowest I would ever go for basketball), then I could run f4.
You should be able to find the first gen of the 70-200 f2.8 lens pretty cheaply I would think, and that is a good lens to start with for sports. The MKII just has better micro-contrast and captures a bit more detail, but it is not necessary to buy that for sports shooting. You don't need IS either considering your shutter speeds will be fast enough to combat your hand movement. IS will just kill your battery too during a 2-3 hour sporting event, I leave mine off.
Elementary gym (new gym with newer lighting)... I was using the 5D3 and 70-200 2.8 for these and swung from ISO 8000 to 16000.
Pro sports allows me more latitude.