Advice on upgrading from a point & shoot?

[mom_boyz2]

I'm really interested in learning about photography but don't know where to start. I really want sharp quality images of my kids. The problem with my P&S is that it takes way too long to focus, so my kids are gone and out of the shot before I can get a pic. Also I want the ability to learn and try manual settings, all the auto settings of the camera I have give me blurry, blown out pictures. What camera should I start with? Is SLR the way to go or would a Bridge camera do what I'm wanting? TIA for any advice or info!!!

*currently using a Nikon coolpix S203 after my Sony Cybershot DSC-S50 screen died after 5 years of LOTS of love!