For most planning a trip to another country, the focus is usually on the big cities of that country. London in England, Rome and Venice and Naples in Italy, Paris in France, etc. These cities are famous tourists hubs because of their historical monuments, museums, parks, etc. But there is only so much you can learn from a country's big cities. The majority of the people you are interacting with are tourists just like yourself.
If you want to fully immerse yourself in the culture of a country you need to get out of the big cities and travel to the less commercial parts of the country. In these parts you get to interact with true locals. Maybe they'll share with you some of the specialty foods of that particular region or what the best restaurants in town are.
A big part of travel is food. Not only does each country have a different cuisine, but so does each region of a country. In touristy cities, you find that everything is much more 'Americanized.' Therefore you may not be as inclined to try the cuisine of the country because there are restaurants with more similar meal options to those from home. You'll find that once you get out of the city the cuisine is much more authentic. Smaller towns are much less 'Americanized,' therefore you are somewhat "forced" to eat the traditional food, which you'll often find is even better than what you usually eat.
The monuments located in the big cities of different countries are iconic but there are thousands of hidden gems located all over a country that so many people have never heard of. I sometimes think that they are even more breathtaking. They are also not flooded with thousands of tourists.
Of course visiting some of the world's most famous cities is necessary when traveling. Take a ride on the London eye. Or a picture "holding" the Leaning Tower of Pisa. Or even climb to the top of the Eiffel Tower. But do not limit your visit to just the big cities because you only get a small taste of the country that way. It may be unsettling at first, but get off the beaten path and go to a small town no one has heard of.
You will meet incredibly interesting people, learn the true way of life, and learn so much more about the culture than you could in any museum.