Oversimplifying greatly, you have three categories to choose from: Tenkara, traditional flyfishing, and spin fishing. Which you select should turn on where you'll be fishing, why you'll be fishing, budget, and personal preference.
Flyfishing means a rod, reel, line, flies, floatant, a tool or two, and some tippet material. Many rods come in sections (5, 6, even 7) that can be assembled once you reach the stream, and there are a few telescoping rods available too. You'll need some kind of protective carrier for the rod, such as the metal tube many come in or a homemade PVC tube. The most elaborate and most expensive option but in my opinion the most fun by far. LL Bean and Orvis have really good starter kits, and you can save money by looking for travel rod and reel set-ups at Cabellas, Wal-Mart, or any general sporting goods store. Warning: flyfishing is addictive and can become a very expensive hobby.
Tenkara is a Japanese method of flyfishing that uses a telescoping rod and line only - no reel. Because it takes up the least space and weighs the least it is the method of choice for UL and SUL backpackers. Plenty of information on other threads on BPL or at www.tenkarausa.com If you're only going to fish when hiking it's probably your most expensive option.
Spin fishing makes use of a telescoping rod and a simpler reel, with a weighted jig instead of a fly (although you could use a fly). Spin rigs are readily available in trail or travel models for easy packing, and are much less expensive than either of the two flyfishing choices. It's also the most versatile means of fishing as it requires no back cast, allows for longer casts and thus greater coverage, and generally fishes underwater (this last will almost always get you more fish).
If you are fishing for food, you'll do best with a spinning rig. I think flyfishing is more fun and requires more skill, and I release all fish I catch unless I plan to eat them then and there. But find out what you like. Once you know what you like and have an itinerary, check with local outfitters or fly shops about fishing where you'll be going. I wouldn't worry about endangered species, as you'll be informed of forbidden targets when you get your license.
Tight lines, Richard