Start-Small, get technique over technology.
Tele-photo's are a bit of a one trick-dog, especially for a beginner, they delver 'instant impact' from the narrow angle of view, cropping out clutter, and making the subject big in the frame. At the same time, long focus ranges, tend to slim Depth-of-Field, so back-grounds get out-of-focus, easily, again disassociating the subject from the setting adding 'impact'....
'Sports' is also pretty vague.. on the one hand you have something like Ice-Hockey.. a tiny little puck being knocked about at near light speed, often in poorly lit ice-rink, usually with horribly hash and contrasty lighing, and the little niggle, basic scene is far from average exposure.... that is probably about as challenging as it gets I imagine. At the other end, you have the old-folks-home lawn-bowls legue.... Nice evenly lit outdoor setting, and 'balls' moving with the velocity of a glacier....
BUT.. take ANY 'sport' and start racking in and filling the frame with 'face'.... a pained expression of conentration and exhaustion of some-one dripping sweat.. a-n-d... what 'sport' are they playing? After initial impact has grabbed a viewers attention, what's left?
Racking 'out' and including 'more' scene from LESS zoom, is where you gve your pictures 'interest' to make sense and explain that 'impact'; where you provide the 'context' to show the viewer, what sport they are playing, and why they are pulling that face, and start 'adding' interest in what you don't crop from the frame.....
CONTEXT IS ALL!! Composition, Composition, COMPOSITION!
Then Technique over Technology.
Shooting sports SO much is in the 'know-how' and not just in the know-how of the camera gear, but of the sport itself... just like bird watching, you have to know the subject to shoot it; where will the bird or ball be? What are they doing? What do you need to include in the frame to explain what you are looking at?
BIG zoom now starts to become an impediment to the beginner; once a few big impact shots have grabbed attention, everything else starts to disappoint, as the big zoom 'looses' so much interest for that impact...
Meanwhile they are hard to use! No, not use 'well'.. at all! No point having frame filling 'reach' if you are most often filling frame with back-ground, not subject! You can pretty much cover half a football pitch with a standard angle lens (35mm aprox on DX) from the side-lines; you can capture the entire scene. Go up to 70mm on DX and now you are getting 'in' on individual players or small groups in the middle of the field.. but try tracking them as they start to 'run'....
As a generalist, my interest in 'sport' has tended to be of the motor veriety; cars and motorbikes may move pretty quick, but they also tend to be reasonably large.. still challenging to not chop in half, or get dissapearing out of the frame with even not 'that' much zoom. School-Sports-Day? If they still let you take photo's of kids without calling the cops(?!?) Again context is all; I could take pictures of my kids running around all day long, anywhere, any time... what tells viewer that was when they won the green ribbon for effort that makes THAT moment 'special'? B-U-T, on 'full-frame' film cameras, 70-210 was as much as I ever used, and even then, in that sort of situation, 28-80 was probably more use, going 'wide' getting context, or getting close to the action BY getting close to the action, and sitting on the finish line or getting down on the grass next to the high-jump or whatever....
On DX Digital; I have the kit 18-55 and the 55-300. That 55-300 isn't a fantastic lens, it's cheap reach, B-U-T still 'useful', a-n-d, as far as the reach goes, its probably TOO much reach for a lot of stations; and at a couple of motor-bike events, its been the 'cross-over' between 18-55 and 55-300 that has had me vexed, and to get the framing I want in that region, 'really' I wanted an 18-140ish super-zoom to save having to swap lenses in a hurry; and I have shot SO LITTLE over 200mm, the extra up top really has been a bit of a waste... as said, with more reach in the bag for film, I rarely reached for it, so no surprise really I don't use it on digital, where even 200mm gives the equivalent framing of a 300mm thanks to the crop-factor...
Which is to beg a suggestion.... 200m on DX is probably more than enough 'zoom' for most sports and certainly most beginners. You are pondering 70-200 lenses so that added reach is already discounted, which begs the suggestion, that IF you went 70-200, the likely 'lack' you 'may' find, if after discovering the one-trick-dog of big impact frame filling zoom, IS the 'context' from the wider end, where 70mm is going to be the pinch-point, and having that extra beneath it in the 55-70 area, which is so often far more useful, starts to become a silent failing... likely to either steer you to use the kit 18-55 more often and grumble you don't get the 'impact' or reach for the big-reach zoom, and grumble you don't get the context... or the subject!
My reccomend then, IS for the 'cheap' now under £100 brand new and should be under £50 2nd hand Nik-Kit 55-200... or possibly an 18-1X0 extended range kit zoom.
The wider end of the zoom, will give you that more oft used Field-of-View for 'context'; meanwhile it will also, IF you use it make tracking sujects before racking easier, to keep them in the frame. Should also dscourage over-using too much 'zoom' and loosing context and interest.
Meanwhile, that 55-200 has HUGE advantage in its favor of being SO 'cheap'.. it gets you into that higher reach range, to see what you can do and start evolving technique cheaply and with the added wide-end, easily, NOT making life quite 'so' hard for yourself. So WHAT it doesn't resolve quite so many lines per inch! Or its 'Focus Speed' isn't quite up to pro expectations.... TECHNIQUE... Before AF we worked with what we got and used manual focus! Starting out, these sort of qualities are NOT going to be the maker-or-breaker of your shots! And £50 gets you into the game, gets you learning, and WHEN you have acquired the technique to exploit it, THEN might be time to ponder the upgrade to something 'better'... and you'd have cash in the bank to do so... but there Here and now is to start small, and learn to get the most of the gear, rather than expecting the gear to deliver the results for you..
On which basis, 'what lens' tends to a point of insignificant semantics.. you could hand me a professional tennis racket, wont make me win Wimbledon when I cant serve a ruddy ball!! I would need a racket for sure, and probably not a table-tennis paddle, but beyond that? TECHNIQUE over TECHNOLOGY! Don't sweat the small stuff! Get a lens in the right region, and go gt the craft.