My partner is typically my travel companion but I don't cancel trips just because she can't join me on the adventure. I think it mostly depends on your comfort level when doing things by yourself. I've been backpacking alone for up to 4 days in the Cascade Mountains in Washington State. I've taken trips to Holland, Scotland, Spain, and Ireland alone. I'm very comfortable when traveling outside of my own country and actually love the time to myself. You only have to consider what it is that YOU want do and what YOU want to see. One of my favorite things to do in a new place is find a dive bar or local pub, sit at the bar, and start asking the bartender what he/she enjoys doing. You'll find the best food and drinks that way.
As Politesse says, at the very least, make sure someone knows your plans and a basic itinerary. If traveling to another country I always try not to act like a tourist even though that can't avoided from time to time. If you look lost or confused it could open you up to being taken advantage of. I think it helps that I'm a 6 foot tall, 190lb, heavily tattooed male with a shaved head and beard, and I certainly don't act "American" when in foreign countries.
Be respectful of the culture wherever you go and show a real interest in learning what the people of that place enjoy and it will be much more fun than just doing touristy site-seeing stuff.