The bad news is.. there is no shoe that does it all; casual shoes look best with casual clothes, hiking shoes look best with hiking clothes, dress shoes look best with nice pants, etc.; it's one of the most annoying and frustrating things when it comes to minimalist packing for long-term travel...
However, the good news is.... no one actually cares what you look like when you're traveling.. no one but you that is... But you're in a different country... whether it's by the clothes on your back, your hair color, your skin tone or the constant look of confusion on your face... you already stand out... so stop worrying about bringing the right shoe for every occasion! You are literally the only person that will notice.
Your best bet is to find the most comfortable, neutral looking, multi-functional shoes available; because you'll be wearing them ALL day, EVERY day.. and for every occasion.
I wear one pair of black, unmarked, low-profile trail-running/hiking shoes for everything.. they look good whether you're wearing shorts, pants or workout shorts. Shoes are a highly personal decision so I won't make recommendations other than that they should be very comfortable and they should be durable enough to withstand the rigors of wearing them constantly in different environments.
I highly recommend not skimping on this one aspect of your attire, they're low-risk in terms of losing them, and the right (or wrong) shoes can make or break your trip.
I love the boa lacing system for shoes, Saucony makes a nice pair, but I opted for the more expensive, higher-quality Salomon Wildcross GTX. Just search boa lacing trail running shoes on Amazon and find something you like.
I also highly recommend shoe inserts that can be replaced as you wear through them - these can be found literally anywhere in the world and they give your shoes a bit more life.
I do pack a pair of barefoot sandals for beach days and hostel hangs, they take up almost zero space and can be attached to the outside of your bag when traveling (or packed inside, your choice).
I went with the Xero brand based on their ratings and reviews, they are extremely thin (by design) and NOT the most comfortable, but they take up the least amount of space and they are very durable, I've had mine for well over two years and they're still in good shape.
Side note: People WILL comment on them, "normal" people think they're weird and/or ugly, while "travelers" will give you a nod of knowing/approval.. it's up to you if you can handle the sass, I think they're great conversation starters. :)