Kids lowlight indoor events mostly - Band/choir concerts, dance recitals,videos

I am essentially a beginner.  I'd like to  try to video my kids events and pull pictures from the video.  I've been told i'd end up with nicer pictures.  I'm sure if I was more knowledgeable about photography, that would be helpful too!  I really struggle getting a nice picture during b-ball games and concerts.  My current camera is a nicer point and shoot that I inherited from my mom, there was no manual with it anymore.  So, it's time for me to make an investment - without breaking the bank.