Yes bigger lenses let in more light, but the measure of how much light it lets in is the f-stop, i.e. the aperture.
So if a lens has a maximum aperture of f2.8 it makes no difference whether the filter size is 50mm or 80mm as to how much light it lets in. f2.8 tells you that. You might as well worry about what colour the writing on the lens is.
How fast the lens focusses is a factor of a number of variables, it depends on the focus motor used - Nikon, Canon (USM) and Sigma (HSM) all have special designations for their fast & silent focus motors - not all lenses have them. It also depends on the AF sensors and processors in the camera, the more expensive cameras have more expensive sensors and more processing power.
The word "fast" when applied to a lens doesn't usually refer to the speed of the focus motor however - it usually refers to the maximum aperture. A "fast" lens is one that has a large maximum aperture. So f2 is twice as "fast" as f2.8 which is twice as "fast" as f4, etc.
So for shooting sports in dimly-lit environments you need a "fast" lens - i.e. one with a wide maximum aperture. Of course shooting sports often demands a fast-focussing lens too.
Just a warning really - the kind of shooting you are proposing requires very expensive equipment to get good results. That's not to say you can't get some good bang for your buck (John and Mark are giving you the goods on that) but just to set some realistic expectations.
You would like to see "pro" level results (wouldn't we all) but you are going into conditions where even the most expensive pro equipment would struggle. With a modest budget and a few careful purchases you will get results far better than you ever could with the P&S, but you're not going to get very many shots that look like the ones in the magazines.