The Survivalist Challenges have you running around collecting various types of herbs. These just start automatically after a few early missions, after which you'll start seeing "An herb you need to collect is nearby" or similar while running around Mordor. This is your hint that one of the herbs that you need for this challenge is nearby.
There are ten different challenges in all, and each challenge has you collecting so many bits of different types of herbs. By looking at your map, you can see what kind you need to collect for your challenge in the upper left. When you hover over a region, you'll also see a list of all the herbs that appear in that region in the upper left corner. Just remember that mushrooms tend to appear more frequently in caves.
Generally the herbs aren't too difficult to find, and they ALWAYS spawn in the spots. If you find an herb that you need somewhere, make note of where. If you have trouble finding it, you can fast travel to the nearest Forge Tower and harvest it. This can be done even if the herb is right next to the tower, allowing you to quickly harvest a plant several times in just a minute or two.
Keep these herbs in mind while running about doing other things and you should manage to finish most of the challenges before you finish the game.
Please note that it may not be possible to finish all of these challenges until after you open up the second area of the game by completing story missions. According to CrucialVelocity (thanks!), it appears that you can only get through the first seven challenges in the first area of the game.