I am also from the US, we will be there in December. From what I have been reading currently, you do not need a covid test to go to France or UK (at least England). As long as you are vaccinated, you will just have to have a covid test scheduled for your 2nd day you are in England. If you are not there on the 2nd day, you won’t need to take it. But if you are you have to take it. So if you are starting in England, your 2nd day being there you will have to take a test. And if you go to France and back to England, you will have to take a test again once in England on your 2nd day. You have to have those tests booked before going to England. It may have changed again. All info is found on the countries health webpages!

For the EU (and Switzerland/Iceland/Norway but not UK), a good starting point is reopen.europa.eu.

I have gone from France to Switzerland without incident. I did get a Swiss COVID QR code online but no one ever checked it. I showed my US CDC documentation at hotel and restaurants. I have the French QR code and it was accepted in Italy and Spain. 

The British government has a website with email alerts for their ever-changing rules. Recently they have been opening up more and more. However, my most recent understanding is that even fully vaccinated passengers need to take a covid test 2 days after arriving in UK (and this needs to be planned before you come to UK). I am not sure if they still require a covid test before coming for vaccinated people too - which is what they were doing a month or two ago. 

I believe you can get info here :

https://www.gov.uk/coronavirus

Here is the UK Link

https://www.gov.uk/guidance/travel-to-england-from-another-country-during-coronavirus-covid-19