You have to travel far to see close. I heard this quote many times since a kid, but understood the true meaning only when I actually went far. And by this I don’t even mean physically travelling to the other side of the globe (which I also did). Even in your own home city you can explore something very far from your everyday life.
We are used to repeat our days. We have habits in everything, the route to work as well as which milk we choose in the supermarket next door. Even a little change can open your eyes, in these examples changing the car to walking one morning (meaning another route) or going to another supermarket that might not have the same milk brand.
Anyone may start exploring physically close but travel very far in the mind. Of course, abroad it’s more likely that several different things hit you that you’ve never thought of before. A holiday to the beach resort might not be the most culture shocking choice, since it is designed for you to feel EXACTLY as at home, most of the times also the travel guide speaks your language. If the purpose of the trip is to relax and have everything as easy as possible, ready package vacation is the one to choose. In this case though, the culture of the country doesn’t really open to you.
How to explore a new culture then? In the end, it takes very little effort. Most important is to keep your mind open, give a possibility for the new ways of living introduce themselves to you. Meeting the locals is a very good start, also when visiting your neighbour city in your own country. There’s always someone who is ready to show you the insights of their culture. Locals are likely to bring you to places that you otherwise wouldn’t find, make you taste something you’ve never heard of, and teach you a part of their history and beliefs.
Listening, hearing, seeing and feeling this new world will make you compare it to your home culture. What do I like better here, what I prefer at home? What could we learn from each other? You might learn a new language, or improve one. Interaction with anything different from you will give you so much, make you grow and think. It will make you richer.
Here I have some examples what kinds of things you could explore. All the pictures are from my travel archives of past years.
Exploring the miracles of nature, temperature changes, different landscapes and extreme experiences in it, getting to know the history and art, changing points of view and interacting through other visions and thoughts, tasting new flavours and sensing the vibes:

Amazing view of Mediterranean sea. I was introduced to Sicilian culture by a local family, I ate fresh melons from their backyard, improved my Italian and explored the history of the area.
Erice, Sicily, Italy / August 2012

Freezing moments with -30C degrees, glittery snow in the air. My home but always an experience to feel it.
Jyväskylä, Finland / December 2012

White river rafting – feeling the great power of the nature, understanding our mortality, living full with adrenaline.
Rotorua, New Zealand / March 2011

Going back in time, feeling the ancient. I found this place to visit thanks to my Sicilian friend.
Segesta, Sicily, Italy / August 2012

Fresh provoking art on shirt with the happy owner – me, Spr Fkr
I participated open doors of the museums of modern art in the downtown of Los Angeles and changed points of view with a local, talented artist.
Los Angeles, California, USA / Februay 2011

Interacting with the visions of the greatest artists of all times in Louvre museum. I have visited several times, but I never get enough of the master pieces in it.
Paris, France / December 2011

Delicious bruschetta bread after a burning day by Mediterranean sea. ‘On a holiday’
Terracina, Italy / August 2011

A very special flavoured liqueur from Finland, tastes similar to liquorice, just saltier. Definitely a must! A ‘little-piece-of-home’ present from a lovely friend visiting me this weekend.
Frankfurt Am Main, Germany / February 2013

Vibes of a lake in the silence of a nightless night. In the middle of all chaos and experiences it is important to stop and reflect.
Keuruu, Finland / June 2008