First, no need to apologize for being new. We all started out that way.
If they look at me and smile I want to be able to capture their expression immediately, and not have a shutter lag of 5 seconds.
That actually sounds more like a combination of auto-focus time (possibly with some additional indoor/low light delay thrown in) and auto-picture mode time, not shutter lag. This ties into what you said about wanting to "learn on the camera" in that, it would probably help to reduce that time if you learned to set more things manually so the camera wouldn't have to take as long trying to figure things out. To that end, most point-and-shoots in the $300-500 range have a manual mode, so you don't necessarily have to jump all the way up to a DSLR (or MILC, Mirrorless Interchangeable Lens Camera, like the G5). You might even want to check if your current camera has a manual (or at least semi-manual -- Shutter and/or Aperture Priority) mode.
In any case, one of the simplest ways to reduce AF (auto-focus) time is to pre-focus. Most cameras allow you to do this with a half-press of the shutter button. To go back to the playground slide scenario, as your kid climbs up the steps, point the camera at the bottom of the slide and half-press the shutter button (and keep it half-pressed). Then, when your kid reaches the bottom of the slide, fully depress the shutter. This should reduce the lag to tenths of a second, even with a point-and-shoot.
Also, when taking candid pics of people, you can use Shutter Priority mode with a shutter speed of 1/60 (or faster, depending on how fast the person is moving. e.g. in the slide example, you may want to use a faster shutter speed of 1/120, while soccer would be faster again, so maybe 1/250).
And those two "tricks" are really just the tip of the iceberg. (And both techniques will transfer over to any DSLR you buy in the future.)
Personally, I think you'd be better served learning these things on a P&S (point-and-shoot) (like the FZ200 or again, maybe even your current camera). And then later, when you know for sure that you're hitting the limits of the camera's performance, and you know for sure that you need to spend a lot more money, and you have a clearer idea of what you need, then upgrade to an interchangeable lens system.
My 2¢ - Mark