@robertw10325
Visas are (usually) an expensive, time-consuming aggravation, residence visas even more so. (Similarly, related incidental such as long-term rentals, private health insurance, local jobs.)
For a relatively short period abroad (6-12 months), I suggest you'd be far better off to avoid them! I like zero hoops, and I've never bothered unless I want to relocate for several years or I've been offered a "proper" job. Americans have visa-free travel to most of the world, typically for up to 90 days (up to 180 days for UK). This means you can easily spend several months in each of several countries.
Some will let you leave and re-enter at 90 days (the "visa run"), but most of Europe allows up to 90 days in 180 (i.e a proper break before returning). Additonally, the EU includes the Schengen Zone, so your 90 in 180 limit applies to all Schengen countries. The non-Schengen countries are non-EU countries (UK, Serbia, Albania, Turkey, Dubai and many more) and EU countries outside Schengen (Ireland, Cyprus).
So you can't do 3 months in Spain, followed by 3 months in France (its neighbour), as they're both Schengen. But you can do 3 months in France, then pop over to UK on the ferry and spend 3 months (or more) there, then get a cheap Ryanair/Wizzair flight to Spain and spend 3 months there.
With these shorter visits, you should be able to have a good travel insurance policy cover you for any emergencies, rather than buying far more expensive private health insurance.
Similarly, finding a place to rent is a time-consuming exercise, and you're paying for an empty apartment when you're travelling. Sure, short-stay rooms (via Airbnb or Booking) are a bit more expensive, but they're much easier, and much more flexible.
It's not legal to work when you're on a tourist visa or a tourist visa-exception, but many remote workers/digital nomads do exactly this for years, across many countries. Tik-tok documenting, optional.
Working in a language school is a different issue, and I would not recommend it (and not just from a work permit perspective). You might find one that will do cash-in-hand, but I think most will want legal workers these days. If you have teaching experience, I reckon it's better to do it online so that it's something you can do wherever you are, without a work visa.
Barcelona is a great city. I like it a lot and I visited the city and the nearby coast (the beautiful Costa Brava) often when I lived nearby (Andorra). But it's very expensive, and I'm not sure I'd want to stay there for more than a month, and I certainly wouldn't pick it as my European base. If you really want a Spanish residence visa, I reckon the NLV (No Lucrativa Visa) is the easiest, provided you have either pension (or other passive) income or 30k euros in savings. Otherwise, as suggested, the DNV.