Love this idea. it's a plan of mine as well. I have done some shorter rentals 1-2 weeks in Argentina, Japan and Spain. Early this year was going to be my 1st try and more of a long term and I was going to do 3-4 weeks each in 2 different areas of Thailand. Covid put an end to that but I'm going to look to the future to do it
So to answer your question a bit, I don't have specific places/apartments to advise but cities or places I'd consider to start Tokyo, Shanghai, Thailand, India (people love it or hate it), Spain (Malaga or the cost), Sicily, I haven't actual been but this is a place I'd like to do a month, same with Crete. Also I'd suggest Argentina for a couple of months, in a couple of places.. The list goes on and on, of course in some of the larger places you with have access to good public transport (like all of Japan, Shanghai, Buenos Aires, etc) and others are cheap enough that you can get around inexpensively with taxi's uber or even local buses if you're open to that.