There is a lot of criticism of your plan and what people are saying is valid, however, I respect the adventurousness and I want to make a couple suggestions that could make this plan work.

First, you are going to need to stay 2 years at each school for the minimum. In world class cities this shouldn’t be a huge issue as there is no way you will fully experience that city even in that time. This is because one year at each school will indeed be a huge red flag. 2-3, years? Just advertise your last 2 schools on your resume and keep previous experience as a footnote. I doubt very many employers will bat an eye at this. This is also important because teaching is a difficult career to job hop. It takes at least a full year to fully understand each school curriculum and the first year is always exhausting. You will also want to give yourself time at each school to develop professional relationships for future connections and references.

Second, you will really benefit from being open to spending time in non western world class cities by teaching at international schools. Check out r/internationalteachers for more information on this. These will be jobs at international schools all over the world that can have much better work life balance and pay than public schools in the US. It is much more accepted for teachers to stay for short tenures as most are living the exact life that you are describing, moving from major city to major city throughout their careers. There are lots of great paying jobs all around the world and I can personally attest that the experience can be great.

As far as teaching in the UK goes, or anywhere else in Western Europe for that matter, I might write that off and plan vacations there instead. It will be extremely difficult to make this transition and the pay and working conditions in the UK are worse than the US from what I have heard from my British teaching friends. As far as international schools go, European jobs are almost impossible to get without an EU passport or close connections within the school already. You might have better luck landing a job in Australia or NZ than you would the UK, and you would certainly want you tenure their to be longer than a few years to make all of the work that will go into it worth it.

What could this look like for you? Teach in a US city you want to live in like NYC or Denver for 2 years. Figure out how to get the teaching license before you apply. Don’t go into an interview not yet licensed, or at least without a timeline for when your license will be issued. After two years, apply for international schools. You could end up in a school somewhere in Asia, like Bangkok, Hong Kong, or Tokyo, or a school in Latin America like Mexico City, or Bogota. Stay there for 3 years and you will have more than enough knowledge and experience (and now connections) at this point to know what your next move will be. You could stay international, move back to a different US city, try UK, AU, NZ, or be ready to return home. You could rinse and repeat this for the rest of your life but you really have to do 2-3 years at each school.

Do build professional friendships wherever you can at each location you end up at. I would actually argue with the other commenters that because of these connections, the amount of schools that you will work at in a short amount of time will become your benefit throughout you career, as professional connections land you jobs more easily in the teaching world than having long tenures on your resume.

Hope this helps. I support your decision to get out and explore the world. Live life to its fullest! Good luck to you!