It really depends on the sport that you're shooting. Some sports let you get fairly close (e.g. hockey, curling, basketball); other sports require you to use long lenses at a distance (baseball, cricket, football/soccer, gridiron football).

Fast lenses are best because of the lighting conditions in some sports, especially indoor sports or games/matches played at night under stadium lighting. Modern DSLRs give you more room for error on the ISO side of things than was the case in the film days, but a fast aperture will still help you shoot isolate the subject against the background, and let you use faster shutter speeds with higher quality.

Knowing the sport well, and knowing where to anticipate action, will help you considerably.

It's been awhile since I've shot curling or hockey, but when I did (as an amateur) I used a 35-135/3.5-4.5 (on 35mm film/full frame) and it was a pretty successful endeavour. On the other hand, with gridiron football I used much longer lenses. My 80-200/2.8 was my most useful lens (I lacked anything longer that was as fast) but I also got a lot of good use out of a modest 75-300/4.5-5.6. In fact, my best football shot was shot with this lens, at the nearer end of the range, as a running back ran toward me on the sideline. (Luckily he turned the corner. :) )

Using what you have, and learning its limitations, will teach you more about what lens to add to your stable than any advice we have. Go experiment and try, see how things turn out, and figure out what limitations you're running into - that will tell you what lens to get next.