I drink a lot of water. I don't even leave the house without my water bottle (and a snack! I am a real life Mary Poppins. I also carry a book and a notebook) and I can't work without a tall glass of ice cold water. Of course, in the excitement of traveling, it can get a little harder to keep track of your water intake/hydration status. Being a little thirsty when you are out and about exploring isn't usually more than an annoyance that can be quickly taken care of, but there are a few instances when you should definitely focus on your water consumption.
The other time you should really be aware of your water intake is when you are hiking or are visiting someplace really hot/desert. The National Park Service states that for every hour you are hiking in the sun, you lose about .5 to 1 quart of fluids. Dehydration can lead to heat exhaustion. Another thing to keep in mind is that when you are sweating, you lose electrolytes. Drinking too much water and not getting enough electrolytes can lead to Hyponatremia (water intoxication) and it can be serious. So along with the water, it is also essential to eat salty snacks or drink an electrolyte drink. You can read more here.
Usually when I am traveling, I like to buy a pack of plastic water bottles and at least 2 large jugs of water from any supermarket as soon as I arrive. Just store them in the back of your car or in your hotel room and then fill them up in the morning. That way I always know that I have water nearby and I am not buying a bottle of water or paying for hotel water every time I get thirsty. Or you can just take an empty water bottle with you. I just like having the security of multiple water bottles if I finish one.