Second hand is an option, but i would prefer to buy brand new.
The reality of the situation on your budget is really fairly simple. You either buy slow, consumer grade glass (not a great option) or you buy second hand. Your budget in not such that you will likely find a good condition, used 70-200 f2.8 Canon lens. You could look for a used Sigma 70-200 f2.8 in good condition for your budget and if you get lucky you might even find one new.
Sports photography is very demanding on the photographer as well as the gear. I shoot sports and easily have $30,000.00 in glass and not one lens is slower than f2.8. Bare bones basic requirements to cover most sport situations would be along the following lines.
70-200mm f2.8
50mm or 85mm for indoor sports.
Long glass 300mm or 400mm if possible for field sports.
A good lens lineup would be as follows and allow you to cover virtually anything.
Generally for indoor court sport use
35mm f1.4
50mm f1.4
80mm f1.8
100mm f2
70-200mm f2.8 (Probably the workhorse even though it is often on a second body indoors and out)
Field Sports
200mm f2.0 (indoor and some field sports under the lights)
300mm f2.8
400mm f2.8
Additionally you might want to look at an ultra wide. I use a Tokina 11-16 f2.8 on a crop frame 3rd body for unique perspective shots.
A 24-70 f2.8 is useful for pregame/post game and close sideline emotion shots.
If you are serious about this you might want to consider renting lenses for events and start saving your money. Good glass does not come cheap and if taken care of can last a lifetime.