I suggest you look at two things in any camera: size of sensor (bigger better) and the f-number of lens (smaller better).
The Sony RX100 has both features in a very compact camera. In the right hands, one can take very pro looking shots.
The canon sx60x has a big zoom, but tiny sensor and huge f-number that gets worse as you zoom. It will take good photos but probably not any better, based on the specs, to your existing point and shoot.
If you have never used an slr before but want to take it up as a serious hobby, the Nikon 3xxx series is a solid choice for you. The Nikon 3400 is a decent entry DSLR and comes with a set of kit lenses which covers a good zoom range. The f-number on the kit lens is larger, which means they are "slow". In other words that hole in the front is small compared to the length of the lens. Except for some basic primes like a 50 mm, you need to spend a lot of money for really good fast slr lenses. But without good technique, spending a lot money on expensive lenses may still result in crooked, blurry or poorly exposed shots. You can always add new lenses in the future.
If you do get the Nikon 3400, i would suggest carrying the 18-55 only. Going from a simple point and shoot to a dslr is a big leap in bulk and weight when travelling. Lugging the extra 70-300 will be a bigger burden. Seasoned photoraphers can carry and change multiple lenses without a second thought. For a newbie, it can be a real pain. Take the one lens and concentrate on composition and experimenting with the settings, especially the exposure compensation. If you want a closer shot, just zoom with your feet.