Alchemy lets you get certain spell effects without having to cast spells. All you need to do is find ingredients (at least two unless you are a Master of Alchemy) that have the same effect, mix them together (you must have at least a Mortar and Pestle) and *poof* you have a potion. There doesn't even appear to be a skill-check to see whether you are successful or not. Potions with positive effects show up in pink bottles while poisons with negative effects show up in green bottles. As long as a potion contains at least one positive effect, it will show up in a pink bottle and you can apply it to yourself. Click on a pink bottle to drink it or click on a green bottle to put poison on your currently equipped weapon.
Alchemy effects are included in the limit of 4 active effects at any given time. If you find yourself at the point where you're needing to swill several Restore Health or Restore Magicka potions and have no other active effects going, the game will stop you after four potions. You'll need to return to the game screen to allow those four to "officially" take effect before you can use any more potions. Unofficially, you saw your health and/or magicka meters increase, so you've received the benefit, but the game still needs to process the effect. Once you've gone to the main screen, you can reenter your inventory and swill up to four more if you need to.
The trick in Oblivion is that you can't see all of the effects until you have reached the Expert Skill level. This means that Novice Alchemists can only make potions that contain the first effects of the ingredient, Apprentices can only make potions that have the first and second effects, and so forth. In practical terms this means that a Novice cannot make a Potion of Night Eye because there is only one ingredient with that effect under the first effect (Daedroth Teeth) and you must have at least two ingredients. An Apprentice, however, can make such a potion because there are three other ingredients that have Night Eye as second effects (Carrots, Daedra Silk and Viper's Bugloss Leaves), but could not make a Potion of Feather because there is only one ingredient (Flax Seeds) that has Feather as the first or second effect. There is an ingredient that has Feather as the third effect, so a Journeyman could make the potion. And an Expert would have a choice of ingredients, though, because there are three other ingredients with Feather as the fourth effect.
As far as working the skill goes, you get some very small skill improvement by munching on raw ingredients (as well as very briefly getting whatever the first effect of that ingredient is), but you're going to get the best improvements by combining ingredients to make potions. You can capitalize on this by stocking up on ingredients with similar effects before you head in to wherever it is. For example, if you're going to be ducking into an Oblivion Gate, you can bring along matching ingredients (Mandrake Root to go with the Clannfear Claws, Cairn Bolete to go with the Daedra Hearts, etc.), which will save the weight. Claws and Hearts are a couple of pounds each, as compared to 0.5 pounds for a potion.
As a relatively quick way of disposing of some of these ingredients and upping my Alchemy skill, I found that Harrada and Spiddal Stick make a workable poison with a Damage Health effect. Before ducking into any towers, I scour the area to pick up any of these that I can find, whip up a batch of poisons and hot-key them. Then it's into the towers, sneak, whip out the bow, hit the hot key, snipe the unsuspecting victim, duck back out of sight while the poison works its magic and repeat as necessary until I'm either out of poisons or out of unsuspecting victims. Alchemy, Marksman and Sneak all in one fell swoop.
As a quick method of working up the skill, snag all of those food items that you come across and use them to make Restore Fatigue potions. You're going to get positively sick of doing it, but your skill will climb by leaps and bounds and you can make a fairly decent sum selling off the resulting potions.
Be warned, however, that you might want to consider stockpiling some of those Oblivion ingredients because once you have finished the Main Quest, you're going to have an awfully hard time getting to Oblivion to collect them. If you're skilled enough in Conjuration, you can summon the appropriate creature and stomp on it, but Bloodgrass, Spiddal Stick, and Harrada don't grow in Cyrodiil, although they can sometimes be had at the better alchemy shops. It appears that Bloodgrass, at least, will occasionally spill over into Tamriel at the site of some Oblivion gates that you close, but it's a very iffy thing.