Canon EF-S 55-250mm f/4-5.6 IS STM is a great budget lens for outdoor, daytime, sports photography. Canon 70-300L is an awesome lens for outdoor daytime sports, my favorite. I have seen many very good used ones from $1,000 down to $700. I paid $900 for a like new one. Outdoor sports under lights will almost certainly require a 70-200mm f/2.8 lens. They are not cheap, they are not even close to being a budget lens.
IF THERE IS A CHANCE YOUR KIDS WILL BE PLAYING UNDER LIGHTS WHEN THEY GET OLDER, get the bargain 55-250 now and start saving for a 70-200mm f/2.8 lens. A few, very few, youth athletic fields have good enough lighting to be shooting with f/4 or slower and still maintain a fast enough shutter speed to stop action.
Sports photography is not cheap even at the lowest amateur level. My bank account knows all too well. It can be very exciting, rewarding and worth every penny when the subjects are your family.
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
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