Moving to Europe for 1 year , maybe2…..

First find if you can do it by obtaining a long stay visa.

welcome to the forum, loris.garden. There are some forum participants who are long term stay folks, hopefully they can weigh in about the steps you will need to take to extend your stay past the normal tourist visit time period. It would be helpful to clarify your citizenship as requirements depend on that information. Depending on the country you choose to stay in the most, then you can figure out whether and how you can bring your pets along, the requirements vary by country (vaccinations, etc.) and quarantine of your animals on arrival is a definite possibility.

This site lists some options you may want to consider. While the site is probably mostly correct, you'll want to use official government sources for info to develop final plans.
https://www.nomadicmatt.com/travel-blogs/how-to-legally-stay-in-europe-for-more-than-90-days/

When you decide which country you might be able to move to, check for Facebook forums of expats for hints and tips. I understand they can be most helpful. And do be aware that quite a bit of red tape will undoubtedly be involved (proof of financial stability, medical insurance, etc.).

Lots of good advice above about getting a visa so you can establish residency for a year. Each country is so different in their requirements. We stayed in Italy for 18 months and the process was a bit arduous. Also, you may be required to file income tax returns in both countries (US and ???) if you stay over 6 months.

So selecting a possible location might be your best first step. Have you ever visited a country you thought you might want to live in?

Germany is quite handy but views by the sea are all north. Italy worked for us for visiting France, Switzerland, Croatia, and Austria easily. Certainly you can get almost anywhere by train or flight and we visited the UK 4 times, once flying and three times via France and the EuroStar train.

Italy means hot summers even by the sea, the UK means cold winters, especially by the sea.

You can spend up to 90 days out of 180 in the Schengen Zone without a visa, but you dare not overstay. I believe the UK allows up to 180 days without a visa. Maybe you can establish 2, 3, or 4 bases for your year?

BTW, as to pets, we moved two cats to Italy, no problem but a lot of hassle as all certifications of health have to happen in the final 10 days before departure including a Dept of Agriculture verification. There are services that help you for a fee. The airline may or may not allow one person to carry two pets in the cabin, so check that as well.

To add what others have said above- if you are thinking of coming to the UK with animals, it's important to know that the UK does not allow any animals in the cabin on either inbound or outbound flights- they must be in the cargo hold. They are quite strict here with bringing animals into the country as rabies has been eradicated here for many years and they want to keep it that way.

"You can spend up to 90 days out of 180 in the Schengen Zone without a visa, but you dare not overstay."

You might also want to look into reciprocal visa waiver agreements. I'm an Australian, and due to visa waiver agreements we can stay in quite a few European countries- Germany, Spain, Portugal, Netherlands, Australia plus others, for 180 days without a visa. Does the US have something similar in place?

Also, do your parents or grandparents have a European citizenship making you eligible for residency or a long term visa?