I want recommendations of locations and schools which teach foreigners korean. English is my first language. I am an older woman who will be traveling alone...so I also wish a situation which would be conducive to making friends.
I'm not sure I can fully address your question but I can give you some suggestions.
Just to clarify, hangeul is the Korean writing system and it's worth learning that before you arrive. www.koreanfromzero.com has a great book that's free for level 1. You can download it right from the website. Learn hangeul first. It isn't complicated and it'll make everything else (traveling included) much easier.
Depending on how serious you are about learning Korean, you may do better with a language exchange as you'll learn some survival Korean and meet nice people. You can also help them practice their English. If you come to Busan, you can probably post on here again just for that and someone will take you up on it.
You mentioned that you're older.... not sure how old that is. If you're trying to meet other foreigners your age, that could be a challenge. There are quite a lot of us in our 40s, some 50s, but most are younger. In Busan, it's a bit more rare to find retirees as it isn't really on the radar. SE Asia would be a better fit for that.
If you're really set on learning Korean in a language school, I briefly took classes at KLIFF (in Busan) a few years back. Not sure if she's still having classes but it was about US $10 for one hour. Jenna at KLIFF is really helpful at giving you a solid jumpstart in conversational Korean. www.kliff.co.kr --- The website hasn't been updated since 2012 so not sure if she's still in business. Someone else on here may know for sure. I have been away since 2012 and have only recently returned.
There are also free classes put on by the government but I'm not sure of the eligibility requirements for these. I'd take a strong guess that tourists aren't able to take advantage.
All of the info above relates to Busan. You'll probably have more luck in Seoul, especially in the Itaeweon, Jongno, Hongik, and possible Gangnam districts. But can't give you much more on that -- I've only taken university programs which were intensive language courses.
Enjoy your time in Korea and do your best to make it to Busan. This is a great time of year to visit.
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