You might be better off not trying to find one lens that meets both needs (football & basketball). For football, you need length first and speed second. For basketball, you need speed first and length second (IMO).
If all your football games are day games then you aren't dealing with low light and the 55-250 or the 75-300 should work.
For basketball, you give up flexibility with with a prime but you gain speed and sharpness. Personally, I would rather have a lens that can give me a higher shutter speed and freeze the action than a zoom that lets me get closer but with a slow shutter speed. Who wants a close up of a blurry photo?
If I were in your shoes, I would go for a fast prime like the 85mm f/1.8 for basketball(assuming you already have the 50mm f/1.8, since everyone starting out should have one). The 50mm does not focus fast enough for sports (in my experience) and the 85mm is such a sweet lens. The portraits you can get with it as well will just WOW you.
For football, the 55-200 gets good reviews, but if you are serious about photography it might be worth it to bump up to the 70-200 f/4.0, its a very good general purpose mid-length zoom lens.
But when you get to night football games, be prepared to spend some serious money or just sit back and watch the game. For night football you need both length AND speed, which costs some big bucks.
All of the above is based on personal prefference on how I like to shoot. Your milage may vary so be sure to listen to all points of view before you make your decision.