Well if your talking about outdoor photography in good lighting, a decent 70-300 mm 4.5-5.6 would probably do the job nicely. If your talking indoor photography where you'll need a large aperture zoom, well sadly there really aren't any options that are going to work well for that on a limited budget.
Definitely outdoors...do you believe that the 4.5-5.6 aperture would be large enough? Just because I am currently trying to figure out how to shoot water (different post) at the high shutter speeds needed to capture the action and I'm having difficulty getting enough light in my shots (kit lens has similar max aperture) even in outdoor light. I am admittedly new to DSLR's...perhaps I'm trying to relate one situation to another where it's not appropriate? Either way, thank you for your opinion!!
As long as you have good lighting you should be able to get some pretty good action shots even with higher shutter speeds, I routinely shoot at those apertures or sometimes even higher with shutter speeds in the 800-1000 range with good lighting with no issues.
I don't use my 70-300 mm all that often to be honest, my bread and butter lens is a 70-200 mm F-2.8, but sadly they aren't really any "budget" versions, you can sometimes find some used non OEM brands like some of the older Tamron's or Sigma's in the $500 range, but these versions will generally be older versions without any sort of image stabilization so that is something to consider, and of course my guess is $500 is most likely more than you were hoping to spend since you mentioned you were on a budget. But sadly to get a decent zoom with a wide aperture is not a cheap endeavor.
I shoot Nikon myself, so maybe one of the Canon folks might know of a couple of lenses to recommend that I might not be all that familiar with myself.