Capturing family life and events

I am new to using a dedicated camera. I am used to using the phone or tablet, unfortunately I am unable to take adequate pictures of my children's sporting events so I am looking for the best, cheap zoom camera system thatwil also provide bursts of pictures for quick action events such as sliding into first base or scroing the winning goal.

I have looked at every camera and all websites available.. OK, maybe not ALL :)

Last qualification for this camera is to be able ti take adequate portrait pictures for when I am Santa during the Christmas season. 10 feet of range, poor to moderate lighting and squirmy children.