As an EU citizen you can live long term in any other EU country. For less than three months you don't need to do anything (apart form having your passport or National ID with you). For longer terms however you have to either be engaged in economic activity (work there), or have enough resources to not become a burden on the social services of the host Member State. As you said you have a steady income, this point should apply to you. More information
While Switzerland, Iceland, Lichtenstein and Norway are not part of the EU, the above also applies for them. You might to register yourself with local authorities (like in Norway or Iceland), and show them you are self supporting.
As countries outside of the EU don't recognize the EU as one, but as separate countries, it depends on your home country where you might live struggle free for a longer term. While this example is for short stay it highlights the issue clearly: most EU countries are part of the Visa Waiver Program, meaning they can enter the USA (for short term stay) easier, but some are not. For example as a French citizen you can stay visa free in Canada, Peru, Mexico and Panama for 180 days (but require visa for longer stay).
(Note: the question was asked generally about EU citizens in the first place)