The best areas for foreign residents to live? (Tohoku region)

This is a tricky question to answer because it depends on a lot of things.

The Tohoku region tends to be a bit more rural than the rest of Japan, which also necessarily means fewer foreigners, fewer foreign goods and everything that entails.

Having said that, it ultimately depends on what you're looking for. If you prefer the city, you"ve got some options: Sendai (Miyagi Prefecture) is the largest city in Tohoku (and the capital of Miyagi Prefecture) and has all of the related accoutrements; plenty of places to party, plenty of other foreigners and foreign goods.

Yamagata city (Yamagata Prefecture), Niigata city (Niigata Prefecture), Akita city (Akita Prefecture) and Aomori city (Aomori Prefecture) are the respective capitals of their prefectures and as such, while quite a bit smaller than Sendai, have a resident foreign population, shops and nightlife, so it's not a bad option.

If you prefer the countryside and want to mostly get away for foreigners, then basically anyplace else is where you want to go.

Living in Yamagata, my personal preference is Sakata city, which is the 3rd largest city in Yamagata Prefecture and is a port town. As such it has a small but vibrant nightlife (particularly on the weekends, though it's been a bit subdued due to Corona), a small foreign contingent if you get homesick, and is reasonably close to both bigger cities and the deep country, so you get the best of both worlds.