Niantic could stand to put a number of tutorials in Pokémon Go, especially when it comes to the nearby system. It took days of testing to figure it out, but we think we have a handle on it. (Excluding a strange bug.)
In short, tapping nearby button at the bottom right of the main screen will show a list of all the Pokémon in your immediate area. Each one has a track marker to show how far away it is. Those with one paw print underneath are the closest and nearly on top of you, two being relatively close and three meaning they're the farthest away. Keep this menu open, resisting the urge to tap an individual creature to focus on it. Start walking.
Now you're going to be looking at the macro picture here, watching the Pokémon shift as you walk. See how the order changes? That's Pokémon Go showing you how close you're getting to each Pokémon. If you see something you want shift down the list, change your direction. If the Pokémon shifts yet again, return to where you were standing when it was higher on the list, and repeat the process, getting closer and closer to the Pokémon.
One more thing to note: Occasionally Pokémon will up and leave, and there's nothing you can do about it. But following the technique above will help you find the next Pokémon — and help you capture rare Pokémon more consistently.