First, the lens that is on my camera the most is a 50mm f1.4.
That said...I'd look at the Nikon 35mm f1.8 DX lens. It's cheap, it's small, it's really light weight. It will be good in low light. it will be good for landscapes, group shots, and architecture (other than the traditional distortion you get with wide-angle lens). The only negative is that I found I got some chromatic aberration in bright light when shooting at extremes (like f1.8).
For travel photography, I'd definitely get a zoom that goes out to at least 100mm. It's handy for wildlife/creatures, street photography, or capturing details when you just can't get closer (b/c there is a road or canal in between you and the subject). But be clear: most zooms aren't the first lens of choice for landscape (you want a wide angle) and you need to get a professional lens for anything that will be good in low light. For instance, a new 70-200mm f2.8 zoom will probably cost you $900-$2,000 depending upon which one you get.
The kit lens is nice but...it's going to be mediocre in low light. And the 35mm prime will be much sharper.
If I were the purchasing director for you trip, I'd budget for the following:
--35mm f1.8 DX
--a cheap 200mm zoom that is small and lightweight (don't worry about the speed) b/c there will be some shots that a 35mm lens won't work for and you'll mostly be using this lens in daylight, not for lowlight, not for interiors, not for landscape. So speed doesn't matter that much. Instead, portability would be your biggie.
--a tripod that you can carry easily and fit in carry-on baggage. And if you say you don't want to carry a tripod, then get a gorillapod. There will be sunsets/sunrises, low light still shots inside a building where you can't use flash and something to stabilize your camera will be vital.
--a circular polarizer to fit on the 35mm (b/c it will reduce glare and make clouds pop and give you blue skies).
I think you can do all 4 of those items for a total of $400-$600 (depending upon if you get a toprate travel tripod or a gorillapod and what type of zoom you get).