To easily find the plant Nirnroot, find a body of water (in a city or out in the world) and travel along the edge of the water. Nirnroot tends to grow very close to rocks, usually when two or more are set directly next to each other. Since the Nirnroot likes to grow in shady areas, it can also be found in locations like under bridges or next to statues. However, for the most part, you will find it right up against rocks shielding it from the sun. A good place to start would be to go to Weye (just outside the Imperial City) and travel around the river/moat that surrounds the city.
Rather than wander all around Cyrodyl looking for Nirnroot, just put it in the ingredients duplicate chest that you get when you finish the Mage's Guild.