And now for a hunk of rank cheese.
*SPOILER!!* Don't read this if you want the game to retain any semblance of challenge...
Although I doubt you'd be very successful with such a low skill level, alchemy is absolutely the easiest skill to quickly advance. You can reach 100 skill in a matter of minutes, provided alchemy was one of your major or minor skills (which it clearly wasn't in your case).
Here's the deal. Step one is to head off to Balmora. Step two is to locate Nalcarya of White Haven's Fine Alchemist shop, which is right next to the clothier on the far side of town from the silt strider. She's in dire need of resist poison potions, and by the time you're done with her, she'll have thousands...
Step three is to get some decent alchemy equipment, and thankfully, Nalcarya is generous enough to have a grandmaster's mortar and pestle sitting right on her shelf... if only you can get that guard to look somewhere else. You'll need a bit of sneak skill, or perhaps a chameleon spell, but in either case, it's a good idea to get the guard to turn around by standing behind him long enough to get his attention. Wait for him to finish speaking to you, and then waltz back over to the shelf. The guard will stay looking in the other direction, making it much easier to swipe the mortar and pestle.
Step four is to create a large, replenishing stockpile of the ingredients you'll need. Nalcarya has a few ingredients that offer the resist poison effect, but the two cheapest are Guar and Alit hide. She carries ten of each. Buy all of them (she'll give you much better prices if she likes you--try bribing her once you're making a profit). Now return to the barter screen; notice that Nalcarya still has ten portions of Guar and Alit hide, because these ingredients automatically replenish every time you barter with her. Now, sell her your ten portions of the hides. Return to the barter screen once again. She now has
twenty portions of Guar and Alit hide: the original ten that she always has, plus the ten that you sold her. If you were to now buy all of them, and then sell them all back, she'd have forty portions every time you barter with her. See where this is going?
Now simply equip your shiny new mortar and pestle, select the ingredients to mix (start out with groups of 50, and work your way up), and give your index finger a nice workout clicking the "Create" button over and over again.

You'll fail a lot when you're first starting out, but you'll quickly start turning a profit selling her your potions. Nalcarya has a replenishing stockpile of 3000 gold (every 24 hours, that is), which is more than enough to get you started. She's basically an infinite money pump, which you can use to not only rapidly improve your alchemy skill level, but also buy training from several other people all over Balmora.
Once you're strong enough to take on Ordinators, you can say goodbye to Nalcarya and start dealing with Creeper instead. But that's another story entirely.
-Fnord