Lumix is just the name of Panasonic's camera line. The DMCs are point and shoot cameras aimed at consumers where as the GH line is more aimed at photographers. The FZ4 specifically is the top of that line and looks more like a mini DSLR but it still has a super zoom point and shoot lens on it that cannot be taken off. DMCs have really good optics for what they are because of the Leica lenses in them but if your goal is to "learn" photography you probably will grow out of the DMC very quickly and because it doesn't have interchangeable lenses it will struggle to grow with you.
Well, that is what the G3 is An entry level DSLR will run you as much as double that with a lens. You could try looking through the used market though, might be able to get a last generation camera for a good price.
Alternatively there are lots of film cameras that are cheap, but I wouldn't suggest that, you will quickly end up spending more on film and developing even though you might get the camera really cheap.