I'm thinking exactly the same thing as Parts. If you can afford it and have the time, go for the full language-school immersion. You will do and see things that you might never encounter as a casual visitor. Depending on your school, you may be taken on excursions to various cultural events or places of significance. You might have a home stay with a family at some point. This is especially good when there are young children in the house. They are also learning their own language, and they tend to be less inhibited in matters of politeness and protocol, more willing to make jokes, correct you if you make errors, and not take things too seriously.
Studying the language can give some insight into a country's point of view, but there is no substitute for living there and experiencing it for yourself.