I'm new with my camera and I was wondering what size filters to get.
You have to look for the ø symbol in the objective or the number behind the frontal cup of your camera. However the simplest thing to do is to use a ruler and measure from the inner edge of your camera. Take the maximum diameter you'll find and than search for the filter that most fit the measure. If you post a photo with a ruler near the objective I can measure it for you.
There is no thread for a filter in the lens of Lumix LZ20 camera.
Your best option is to hold a filter in front of the lens with your fingers. Another option is to search for a filter that has outer diameter exactly match the outmost part of the lens and fasten the filter with tape onto the lens. Though this latter method will quite likely lead to vignetting in photos when at the wide end of the zoom. Also the old lens cap is perhaps not as tight fit on the filter as it should / too tight fit.