I think that the most authentic way to visit a place is to spend time with it's inhabitants. Couch surfing is obviously affordable, but more importantly, it gives you an instant connexion with welcoming people who also share a passion for travel.

When I was 23, I went as a beginner to a 5-week french immersion camp where I was no longer allowed to speak english. As a personal challenge, I used nothing but my limited, but growing, understanding of the french language. I continued this way for two more years, transferring all of my studies to a french university and I now work as a professor with the same program sharing my own passion for the language.