The first couple of times I travelled a lot. Too much. Moving on every day or two. Seeing the various famous sites to see along the way. It was a great adventure, but I didn't learn that much.
One day I was talking to someone on my daily train commute out of london and she was telling me about a friend of hers that was travelling the world by boat. I thought it sounded amazing, and I always wanted to sail, so I booked myself in to do a leg of the Clipper Ventures [0] yacht race - from Liverpool (England) to Brazil (took about 4 weeks).
When I arrived in Brazil I didn't have a plan at all. I booked in to learn Portuguese while stating with a local family. The 4 weeks I lived there changed the way I look at travelling. I learned about the local culture, made friends, and learned a lot about myself in the process.
In a lot of ways tourism is a much easier route than immersing yourself in a culture. Being totally alone in a country where you can't speak the language is pretty soul expanding.
I've only managed one (big) trip since then (with my now wife) through Central America. We stayed with a family in Nicaragua for a couple of weeks at the start to learn Spanish. We loved it. The hosts are always amazing (we ended up doing it a few times during the trip). Also, it's really cheap. I think we paid $100 / week for accommodation, food and school (for both of us).
So my main bit of advice would be, try to be a local, not a tourist. It's scary, but incredibly rewarding.