Let's start off by talking about equipment. As a long-term traveler (10 years of overseas travel, going on 4 years of non-stop nomadism), I personally travel with a Canon Rebel T4i (DSLR), a Go Pro 4, and an iPhone 7, but I could definitely get away with just the iPhone if I wasn't obsessed with my powerful and versatile 18-200mm zoom lens.
However, what you use to capture your adventures is nowhere near as important as the purposefulness and heart behind the experiences you capture -- and the experiences themselves, of course. I've gotten absolutely incredible photographs over the years, not because of the lens of my DSLR, but because of the subjects I happened to be blessed with and the interesting places I've traveled.
The most important way to make sure you capture your travels the way you want is to be purposeful about it. Make sure you set aside time to document what you're experiencing. Most of the time, I'm traveling slow, so I set aside time to take photos when I first arrive and my first impressions of the place are fresh. Those impressions give me the ability to capture the most unique elements of a new place, the kind of things that strike me as particular to this place are the same things someone looking at my photographs of that place will appreciate: the sombreros the older generations still wear, the crumbling colonial architecture, the cliffs perched over the ocean, the mist that rolls in every morning over the village.
And then I do something else that's very important: I put my camera away. I go out and explore the places I've already documented so I can run around and enjoy it without compromising my in-the-moment experience by trying to set up the perfect shots. Separate out these two aspects of your travels so you take the time to get really good photos and make a special effort to remove any barriers between you and your direct experience of a new place.
Now, when it comes to documenting your experience, that means you need to get in your own photos! Of course traveling with others helps with this because you can all take pictures for each other and swap photos after the trip, but you can still get great shots of yourself in front of those famous world monuments, even if you're flying solo.
If you're on your own, don't be afraid to ask strangers to take your picture. Yes, that means I regularly hand over my $2,000 DSLR or $850 iPhone to Nepalese villagers or fellow tourists on the top of Machu Picchu. It's the only way to get the shot. I'm also especially brazen so I'll ask them to take several shots and even different angles, and nobody seems to mind if I return the favor. If you're shy, you just won't get anything except selfies.