D&D is characterized by encounters. Now, there's no Dungeon Master to sprout encounters in your real life travels, but you can take the Dungeon Mastering into your own hands. In Dungeons and Dragons there are two styles of play: following and leading. Following is when you follow the hints of the Dungeon Master like when there's a crazy old man you keep hearing about, and so you go and find him. Leading is when you create your own objectives. This is like if you decide all on your own to form a political party and then rebel against the king. Now the equivalent of following in real life travelling would be like the everyday tourist who goes to the suggested spots to visit. There is nothing wrong with this. But I'm going to talk about the leading style of traveling. The leading traveler finds their own places to visit and their own things to do, creating more of a special relationship with the area they are visiting. Just how a lot of campaigns of D&D take place over a long amount of time, this travel style is suited for longer visits where you can become engrossed with the new culture.
Step 1: Inquiry
You will be asking a lot of questions. Locals will guide you to where you want to go if you're kind enough to them. Make some friends and you will undoubtedly find an in to an interesting adventure. Finding your next adventure should come from spontaneity as much as possible. You have to be able to adapt and to learn as you go. You obviously can't just walk up to random people and seek out an adventure from them if they are trying to keep to themselves, but you can go to open events and try to talk to as many people as possible. Encounters can contribute to fun in two ways: through learning a story, or through problem solving. Although there will not be a Dungeon Master to craft intricate puzzles for you, you can find for yourself a lot of interesting problems in the world. I don't mean that you should be nosy and try to solve other people's personal problems, but you can help rather when there's open problems that you can contribute to fixing. This usually means that you have to connect with a group that works together towards a common goal. A difference between real life and Dungeons and Dragons is that you won't be doing a lot of the physical problem solving that you do in the game like fighting or finding a way across a room filled with traps. It will be more conversation based, focusing on the development of the group's ideas that you are helping. Look for groups that are open to anyone and that can take general ideas as help. Or you can offer your specialized skills to the group, which would make you a distinct part of the group. If you really want to become immersed in a certain culture, this approach will help because instead of simply seeing the highlights of the culture and just scratching the surface, you will work within the society and temporarily become a part of it.
Step summary: Attend open events that let you interact with and join public groups
Step 2: Collaborating With Locals
Participating in groups can get you deep into a story within the area you are in. The groups you should look for are groups that have an external purpose, or goals that affect things or people outside the group. Thus they can have an effect on their society which means you can too. Joining multiple groups will get you an in-depth understanding of the goals of the society you are staying with. It's like acquiring a quest in Dungeons and Dragons, since you receive the goals of others as your own. You can always suggest new goals that may be aligned with their deeper goals, and thus you create your own goals without overstepping your bounds within the group. Though you may not get into any combat in your travels, you will at least be able to truly say that you went on an adventure.
Step summary: Join multiple groups, help work towards their goals, and give them ideas for other goals they could go after.
Feel free to suggest other ways to travel in a Dungeons and Dragons manner!