I myself made the switch from a Panasonic FZ 15 to the Nikon D50. I do miss the all-in-one convenience and image stabilization of the Panasonic but the increase in image quality of the D50, faster start up time, increased low light ability, etc. more than make up the difference.
Try and get some good glass even if it means buying one lens at a time. The kit lens is actually a good lens from what I've seen. A great lens for a cheap price, ($100), is the Nikon 50mm 1.8 prime lens. It's great for shooting pictures of your kids indoors and out. The f/1.8 aperture allows you to shoot under many lighting conditions and the superlative optics provide incredibly sharp photos.To start this may be a good choice because you would have a little more money for a long lens.
Have you considered a used lens? Below is a link to KEH.com. They sell used equipment and have a good reputation. This is a link to the Sigma 70-300 APO that every one likes so much.
http://www.keh.com/OnLineStore/ProductDetail.aspx
I use a Nikon 70-210 as my long lens. It is a little short compared to the Panasonic's range but it does take beautiful action shots of my son's soccer games. Pictures of birds in the distance are a bit soft but I'm not sure if that's the lens, the photographer or the lack of a good tripod

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I find I must be more careful with the settings on the D50 over the FZ 15 because there are many more options that you can unintentionally change, like white balance or ISO, and find out later your pictures look bad. Some people will recommend you shoot in RAW to correct these mistakes. Right now I'm using JPEG while still getting used to the nuances of the camera. I'm paying special attention to my settings while I'm shooting to minimize my switchology errors. Eventually I plan on switching to RAW but that's in the future. My point is, this is a much more complex instrument than the FZ20, it will reward you will brilliant pictures but you must work to achieve these results.
Enjoy!
Bob