What brand
I usually tell people to buy the best rod that they can afford . However if you are not an avid fisherman, and will not spend much time at it, perhaps look at something cheaper. But for sure, get a rod with a lifetime warranty.
Most companies now offer multi-piece rods. They only weigh 3 to 4 ounces. If I were backpacking, that is what I would take. Easy to carry, less likely to bust in transit. Less likely to get stolen. If money is an issue, look at the Redington Wayfarer, Cabelas catalog on the web, St. Croix. If money is not an issue, look at any of the premium rod builders and the others listed above.. They are all good.
One of the first thing that you should do is go to your local fly shop. They probably have just what you need. For sure they will be able to give you the same advice that you are getting here. Perhaps better!! you are going to need a lot of advice on which line to take, what kind of tippet and leaders, etc. also you will have to carry your flies. You cannot count on being able to buy flies anywhere once you leave the states! You better take EVERYTHING that you will need!