Seeking Your Roots
At some point, probably early on the game, you're going to find a strange glowing plant called "Nirnroot" that makes a weird sound when you stand near it. If you pick up the plant (unlike other flora, you actually take the whole plant and it won't respawn), you'll get a journal entry letting you know that someone would be interested in this plant. Yep, this is one of those "find the quest-giver" quests. The quest has five stages and finding the quest-giver is the first. I highly recommend this quest for low-level characters for a couple of reasons. First off, it's not very difficult, just time-consuming. Secondly, doing it will give you a chance to develop your combat and/or magic skills against opponents that aren't too terribly dangerous or difficult. Thirdly, it will put a lot of locations on your overland map, which will make getting to other quest locations much easier.
Stage One: If you have Nirnroot in your inventory and speak to any alchemist, you'll have a Nirnroot topic that you should follow up on. You'll be pointed to a fellow named Sinderion in Skingrad, but nothing more specific as to his location. Within Skingrad, I've only been able to find two ways of finding Sinderion. You can stumble over him directly by going to the West Weald Inn - the door from the inn's main room shows you his name. Alternatively you can talk to Falana Hlaalu, the alchemist at All Things Alchemical just around the corner from the Inn, who can tell you that he has a lab in the basement of the Inn. Oddly, she greets you by telling you that she's the only alchemist in town. You can also get a chuckle from her by playing along when she asks you about the penalties for necrophilia.
Anyway, once you find Sinderion, he'll ask you to go find more Nirnroot (10, total) for his research. He'll give you an article on Nirnroot and mark a place called Shadeleaf Copse on your map (you'll find three plants there, if you decide to go). Your reward for doing this will be a free sample of the "Elixir of Exploration" that he will make from the Nirnroot you're going to collect. The potion is decent enough, especially for low-level charcters, but the side benefits make this quest more than worthwhile. After you bring the 10 that he wants, he'll send you out for more, so let's just take care of all of this in one fell swoop.
Read the article that Sinderion gave you. Nirnroot grows near water. And that's something that Cyrodiil has in abundance. There are 263 plants in exterior locations (43 others in interior locations) scattered across the map.
Each successive quest stage requires you to collect even more Nirnroot.
If you'd like to just get on with it, head down to the southern coast, as far as you can go and just follow the water (don't forget to check out any islands you can get to - you'll usually find one or two plants on them). If you choose to start on the west bank of the Niben, you'll work your way up and around the shore of Lake Rumare and back down the east bank. If you choose to start on the west bank, it will be just the opposite. Just follow the water's edge down all of the rivers. By the time you get to the opposite coastline, you should have something near 80 Nirnroots in your inventory. Now head for the Imperial City and go around its shoreline to pick up another six to ten plants. Then go to Anvil and follow the coastline between the Strid and Brena Rivers. Don't forget to scour the lands around the cities. If you don't have 100 Nirnroots by this point, you should be awfully close. Go to Leyawiin and Cheydinhal and look along the rivers/canals (edit 05/08 - there are two inside Bravil: one behind the Mages Guild and one behind the house you can buy). When all is said and done, you should have your 100 Nirnroots. If not, head for the Blackwood area, near the border with Black Marsh. There are lots of swampy areas and, consequently, lots of Nirnroots growing around their edges.
If, for some reason, you have to head for a town, just fast travel from a marked location (or mark your own). Take care of business and then pick up where you left off. I can tell you from experience that you're going to learn to hate mud crabs. They have an ungodly detection range and will chase you forever (by which time you've probably run into another). They are apparently Cyrodiil's answer to Morrowind's Cliff Racers.
You should now have dozens of locations on your map and you didn't have to pop into any of them, although you can go back if you'd like. If you can pick the lock, there is one in the Smuggler's Cave behind Castle Anvil - it's usually unoccupied. If you started this off fairly early in the game, you didn't have to fight anything more dangerous than a rat or mud crab or, possibly, a wolf. If you waited until later, then it was probably rats, mudcrabs, wolves, timber wolves, bears, spriggans, imps, will-o-wisps, minotaurs, etc.
Along the way, you also developed your Athleticism, Acrobatics, combat, magic, and other skills and maybe gained a level or two in the process.
Now it's time to go talk to Sinderion. This part will take four full days. Give him the first 10 Nirnroots and wait 24 hours. Pick up your sample of the Weak Elixir of Exploration and give him the next 20. Wait 24 hours and collect your sample of the Moderate Elixir of Exploration and give him the next 30. Wait 24 hours, collect your sample of the Strong Elixir of Exploration and give him the last 40. Wait 24 hours, collect your sample of the Grand Elixir of Exploration and you're done and have gained one point of Fame in addition to the rest of the goodies.
Here's what your potions do:
But wait, that's not all. After he finishes his research, he'll continue to buy Nirnroot from you at 250 gold per 10 that you bring him. So as you stumble across them in your quests and random explorations, remember that there's a little gold waiting for you. If push comes to shove, you CAN use Nirnroots in potions (poisons, actually). It has these effects: