You'll need to protect the lens on FZ20 and there's no point in putting anything thats not up to standard infront of that lens. The first thing you should do before getting any filters is get one of the third party adapters for the FZ's
Check the FZ10 and adapters section on Dr. Shene's website http://www.cs.mtu.edu/~shene/DigiCam/
I got a permaraal PA62 (metal 62mm adapter) and a lot of others have the raynox and Phayee.
For filters your going to need:
Ultra Violet (UV), Neutral Density (ND) and a Linear polarising filters.
Hoya and Tiffen make good quality filters at a reasonable price. The UV filter thats been really recommended is Hoya Super HMC Pro 1 UV. The 62mm version costs about £35 and the 72mm £46.00 from Jacobs and £5-10 on each of those if you get them from Jessops but Jessops will price match.
Here's a previous thread which will help you
Hope that helps