I am unable to locate a list from trusted source (.gov) with a list of countries where GC bearer can travel to without obtaining visa. I know that Canada, Switzerland, Jamaica and Mexico treat LPRs as US citizens. If you have a complete list, let me and us know!
If you are not american citizen, you need visa for schengen states (13 countries in EU, incl france, germany and italy...USD 90 a pop)...most probably, UK also needs a US passport for visa waiver (USD 195 a pop), so does Japan...