For sports / action, you have a few options, depending on how much you want to spend:
Nikon 70-300mm VR AFP f/4.5-6.3 DX ED - You need to update your D5300 firmware to the latest available to enable support for AFP lenses
Nikon 70-300mm VR FX AF-S f/4.5-5.6 G - older lens, look for a used copy
Tamron 18-400mm f/3.5-6.3 Di II VC HLD
Nikon 300mm f/4 AF-S - older lens, look for a used copy, 'extend' with 1.4 teleconverter to 420mm (no VR support)
Nikon 300mm f/4 E PF VR ED AF-S - look for a used copy, 'extend' with 1.4 teleconverter to 420mm
Nikon 500mm f/5.6 PF VR - pretty new lens, used will be difficult to find
The zoom lenses are general purpose including sports, with fast quick AF and VR support and great image quality. They have corresponding non VR versions, make sure you confirm the version you purchase if shopping strictly on price. With your cropped body, they provide a focal length (along with cropping in post) to capture action mid-field for sports like soccer and football.
The 300mm and 500mm prime lenses are
investments in serious sports and wildlife photography for pro-grade lenses that will last forever, where shopping for used versions delivers superior image quality at a relatively modest cost (vs how much more you could spend in this category). The 70-300 lenses will need cropping to get to the details midfield, the 300 primes will deliver a higher quality image before cropping as well as providing options to extend the effective focal length. The 500mm will have the other parents looking at you with new eyes.
For sports / action, you have a few options, depen... (