Can a UK permanent resident travel to Europe without visa?
You do not need a visa for short trips to the EU or countries in the Schengen area if both of the following apply: you're staying for 90 days or less in a 180-day period. you're visiting as a tourist or for certain other reasons.
Citizens of these countries residing in the UK and planning to visit France are required to get a French visa to enter France. The main requirement is that you must have a UK residence permit valid for at least one month beyond the date you plan to leave France / the Schengen Area.
British citizens do not require a visa if spending up to 90 days over a 180 days period in the Schengen area. You may however be asked like any other third country nationals to justify your travel by the Immigration Officer (means of living, address of stay, medical travel insurance…).
You must take your permit with you if you go abroad while you have permission to stay in the United Kingdom. You may have to show the permit to an immigration officer at the border when you leave the United Kingdom.
Applying for a UK to USA Visa can be exhausting and demanding, but when you have a British passport, that's no issue as British citizens are under an American visa-waiver program. If you have a British passport you don't need a specific US Visa for UK citizens to enter the US, you only need an ESTA.
If you would normally require a visa to enter the UK, then you will have to apply for one even if you have a Green Card. Unfortunately, a Green Card does not allow the holder to enter the UK just like it does not allow them to enter the European Union.