A great way to explore a new city is to hop on some public transport, pick a random station or stop, get off and just walk around. Don't worry about getting lost, you can always find your way back. Make sure you have some cash on you for food (preferably small bills/change). I always carry a survival kit just in case, but have only ever needed to use the button compass as my personal navigation system has never worked properly. This activity does have the potential of going horribly wrong so here are a few tips;
Take trains when you can
They are much easier to navigate and the ticket machines are usually easy to figure out. I was once in Guanzhou (China) and a bus driver just started yelling at me. As it turned out, what I thought was a $5 note was in fact a 50c note. To make thins worse, my father was Chinese so by looking at me he probably thought I was just a local trying to rip him off.
Look back when walking
This will help you remember where you came from. In Shenzhen I just kept walking. By the time I wanted to go back I couldn't remember where I had come from and was way out of the train zones. I walked for 5 hours before finding someone that could speak English and point me to a bus back to the city.
Write the address of your hotel down in native characters
This way you can just point at them to a taxi driver. Do the same with airport and other useful places.
Know where the 'dodgy' areas are and avoid them.
One time in New York I caught the subway in the wrong direction and ended up in the Bronx. Luckily it was in the middle of the day. I walked around for a little bit but I'm pretty sure I saw a gang of Crypts at the bus station. I decided not to catch that bus and headed back to the subway.
Start off in the morning and give yourself the whole day
You may just get lunch and go back to the city (or wherever you're staying), or you may get stuck for a while. My planned 'short trip' turned into 36 hours in Malaysia due to infrequent buses. I was later told by a local that the buses only come once or twice a week. That local was the youngest of a traditional Kam Pung (a traditional Malaysian village) family who took me in for the night. This turned out to be one of the most memorable experiences I've had in Malaysia.
For those that aren't so inclined you can stick to taxis to the museum, but for everyone else I highly recommend the 'bus to nowhere' method. The minor mishaps you may have on the way are definitely worth what you may encounter.
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