Seashore is right...basically, that nice shallow focus effect comes from the lens more than the camera, but is also a byproduct of a large sensor like those found on DSLRs. So your best bet for those types of shots will come from a DSLR and a fast or long lens. Now, that doesn't necessarily mean it's the ONLY way - just the best. You may decide you don't need the background to be smooth as syrup - maybe just a little softer and more blurred than the subject - in which case you won't necessarily need a super big aperture lens - even a DSLR with a kit lens might suffice for your needs, and be better than a P&S camera by far. The same goes for capturing action or movement - DSLRs by nature are better than P&S cameras - the lenses can be bought which are faster and brighter as needed for the quick shutter speeds, the focus systems are significantly faster and can track better, and they usually have faster continuous shooting modes.
Note that really any DSLR will be far more capable than your P&S, and even the simplest DSLR has far more ability and controls than you would know how to use or control, but they all function well enough in Auto mode too that you could get one, still shoot with it, get decent results, have greater abilities than you do with a P&S camera, and moreover have a ton of room to learn and grow with it as you do get more skilled.
The question is: do you want a DSLR? They're bigger, heavier, and often end up with multiple lenses to cover all your needs. Some to consider, that have decent 'auto' modes, and which have good prices, and might have some advantages for you to adapt to if not as skilled or knowledgeable about photography or DSLRs:
- Pentax KX - they can be found under $500, have excellent high ISO ability for low light, fast continuous frame rates, and are small and light compared to many other DSLRs...they have plenty of buttons and functions, but could be left in auto and work just fine, giving you time to get to know the camera. This camera has stabilization built into the body, so any attached lens would be stabilized.
- Sony A33 or A55 - very small and lightweight bodies, due to a specialized design with a fixed mirror inside (they're not technically 'DSLRs' but share the identical sensor and speed), can be found from $500-800ish depending on the model and lens choice. These have very good auto modes as well as plenty of manual capabilities as you get better, have the best video focus solution of any DSLR if you intend to use video at all, and for a P&S camera user, they can shoot in 'live view' mode using an electronic viewfinder or the LCD on the back of the camera at full, normal speed - focus, metering, continuous shooting, etc. This is unique with DSLRs, as only Sony versions can do this - all other DSLRs are very slow and limited in live view mode. They are very good at high ISO and very fast continuous shooters as well, at up to 10 frames per second. It also has stabilization in the body, so any attached lens is stabilized.
What else may work for you? Well it might be worth considering one of the new mirrorless large sensor cameras...these cameras have DSLR-sized sensors, but in very compact bodies much like a P&S camera - they take interchangeable lenses, but these are often much smaller and lighter than normal DSLR lenses. They can also be adapted to take full DSLR lenses for those who want to expand, and have full manual controls like a DSLR, but in a simplified menu and button layout required by the tiny bodies. They can equal DSLRs in every image quality statistic, from daylight to low light...the only place where they can't really keep up with DSLRs is in specialized shooting, like sports action, flying birds, etc. For kids moving around a yard, paired with the right lens and the high ISO ability, these cameras could probably handle your needs. These cameras include the Panasonic GH series, the Olympus PEN series, and the Sony NEX series. Many DSLR owners end up with one of these as a second camera, because they are so much smaller and lighter, yet can deliver equal image quality and still offer the photographic control that experienced shooters want...but they also have P&S style auto modes that are usually much more advanced than the auto modes on DSLRs - they can do the stuff P&S cameras can, like smile detection, face detection, automatic image stacking, built in HDR, automatic scene recognition, and more. They go for right around $500, and up depending on the model and lens...these might really be a good mid-way solution for you.
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