If you don't want to change lenses at all, get the Nikon 18-200. It'll set you back something like $600.
This is where you have to think about trade offs and what is important to YOU.
That 18-200mm lens certainly has a wide range of focal lengths covered but it is a slow f/5.6 lens. What if you want faster? What if yo know you almost never shoot using a lens longer than 100mm? If that's you then the 18-200mm is not the right lens for you.
The best think to do is take the camera out right now with the kit 18-55 lens and shoot some photos like the kind you'd take on the trip. Do about 1,000 or so of them. Then look at the result and try and figure out which photos you could not get with the 18-55 and buy the lens that would get those photos.
Whatever you do don't buy some new gear and then use it the first time on vacation. Shoot many, many frames with it before you leave. Yes a lens is technically easy to use, just turn the zoom ring. But learniong how to use it to capture what your eyes see in a photograph is not something you can learn quickly. Shoot 1,000 or so frames before you leave. Look at some of those coffe table photo books and see how the pros did it. You can mostly guess what kind of lens they used. (Almost never a 200mm f/5.6 zoom.)
What lens to bring? The better photographers will all recommend something that is both very wide (less then 18 or 20 mm) and very fast (f/2.8 or better) these guys know that the best images are done when the sun is close to the horizon and they know the golden rule: "get closer, get closer, repeat as required". On the other hand you see the typical vacation snapshooter with a long slow lens using it in harsh daylight and shooting from a distance with a long lens. Which is best depends on your style and intended results.
If you are shooting pictures of people you meet the best lens in my opinion might be the 50mm f/1.4. t is fast enough to isolate a subject and to shoot indoors without flash. But this kind of shooting takes effort you have to ask the subject about photos and take a few to get them used to the camera. and then you might move to where the light and backgrond is better. Again, which lens is best depends on how you will use it and what subjects you are after.
