Want some culture? Go the "sento," or public bath. Not one of those fancy, remodeled ones, but an old, downtrodden one. The best sentos are found in old neighborhoods. Look at the people going in and out. No one should be under the age of 80. These people have been going to the sento their whole lives and probably still live in houses that don't have baths. This is good, because you will see how people use these traditional public baths.
When you walk in, you'll walk into a big room with naked people. This is the changing room, where people will be in various stages of balancing on one leg as they stick the other leg in or out of their slacks. The naked ones fanning themselves are the ones who have just finished soaking in the bath and are so hot, they're about to faint. The fans, provided by the sento, are a form of CPR meant to revive you.
I like to do some warmup activities before getting into the bath, which is another good reason to choose an old sento, because they always have old clunker "misfitness machines" in various stages of disrepair. Think of sentos as the first amusement parks in Japan.