Almost every camera has an auto-mode which should be easy to use. The catch is that they don't always take the best pictures. So it would help if you could tell us which Lumix P&S you currently have, how you're using it (more specifically than "family adventures"--which could be anything from posing with Mickey at Disneyland at noon to zip-lining down Fremont Street in Vegas at night--and "kids playing sports"--which could be a 6-year-old playing T-ball to a 16-year-old playing indoor club volleyball), and what you find frustrating about it. (Some example pics with EXIF data--e.g. posted on Flickr--would be extremely helpful.)
I am always very frustrated with the quality and zoom of the pictures from my Lumix P&S. I want to be able to take pictures of the family and our adventures. I also would like to start taking more nature photography. I know that I want a Compact System (SLR-like) or a DSLR. I cant spend over $600 so I cant spend a lot on the body without it including a lense. Sams Club has the T3 with 2 lenses and a case for $599. I like the idea of the SLR- like ones because they are smaller and lighter and simpler to use. This needs to be simple for me or I wont end up using the camera. I want to be able to have a very good zoom, take videos, and take action shot of the kids playing sports. Even if I go with a compact system one I have no idea which one. I do want one with a view finder which the Lumix have and they are rated well on consumer reports.