Like previously mentioned, you will have a tough time getting what you need with a $1,000 budget. A camera that gives decent results at ISO 6400 and a fast telephoto zoom lens will run $2,000 to $2,500 if they are new.
My experience
Well lighted school gyms and football fields (soccer) at night: ISO 6400, f/2.8 and 1/640 sec. There is a 100% chance of motion blur with any shutter speed below 1/500 sec.
A 70-200 mm lens will cover very well just over 1/2 of a basketball court when sitting behind the baseline. Focal lengths can vary wildly for cheerleaders. The teams can spread so wide an ultra wide lens is needed, 70mm is often too wide unless concentrating on one person.
A 70-200mm lens will cover less than 1/4 of a football or full size soccer field. They are perfect for reduced size youth soccer fields.
Baseball fields: Lighting ranges from impossibly dark to same as football fields. A 70-200mm lens is too short for diagonal shots across the infield. Forget the outfield unless setting up down the foul line and shooting the nearest player.
I love my 70-300mm lens for daytime soccer and baseball (short for outfield.) It is useless at night or inside.
What to do? First, you need a fast lens. I don't see how you get a new camera and fast lens. The Tamron 70-200mm f/2.8 Di LD lens is one of the cheapest. Adorama list it for $549 new. I know nothing about this lens.
Second, a body newer than your 30D with better high (3200+) ISO performance would really help for those action shots in less than ideal light. You have little choice for a camera other than something used like a 50D or 60D. Buying a used camera can be a crap shoot, be careful. A refurbished camera from Canon is a good option when they are on sale, not now.
Canon 7D Mark II w/Canon BG-E16 Battery Grip; Canon EOS 50D w/Canon Battery Grip; Canon SL1; Tokina 12mm - 24mm f/4 PRO DX II; Canon EF 24-105mm f/4L IS; Canon EF 70-200mm f/2.8L IS; Canon 70-300mm f/4-5.6L IS; Sigma 85mm f/1.4 EX DG HSM; Canon EF 300mm f/4L IS; Canon EF-S 18-135mm f/3.5-5.6 IS STM; Canon Extender EF 1.4x II; Canon Extender EF 2x II; Canon Speedlite 430EX II Flash
Image Editing Okay