Hanging Moss + Bear Claw + Giant's Toe = Fortify Heath = Large increase in Alchemy skill. It's also a good way to make coin when you sell it and it...well...fortifies your health.
In Oblivion, yes. In Skyrim, no.
Salmon Roe with Histcarp, Garlic, Chicken Egg, or Nordic Barnacles are good for raising alchemy as well.
If you're not above a little cheating, use http://skyrim.melian.cc/?cmd=cmdSkyrimAlchemyWizard to discover all the effects of your ingredients. Much faster than tasting and/or experimenting.
Then just go through the list of possible potions and make the most expensive ones. Fortunately the gold value is displayed before you create a potion.
Some mod suggestions for an alchemy focused character:
Community Skill Uncapper
http://skyrim.nexusmods.com/mods/1175/?
Prosperous Alchemist
http://skyrim.nexusmods.com/mods/38634/?
Learn Alchemy from recipes
http://skyrim.nexusmods.com/mods/34795/?
Combine potions
http://skyrim.nexusmods.com/mods/15005/?
Field Alchemy
http://skyrim.nexusmods.com/mods/27153/?
Throwing Weapons Redux
http://skyrim.nexusmods.com/mods/36147/?
Farm Everywhere
http://skyrim.nexusmods.com/mods/25655/?
Big Book of Alchemy V2
http://skyrim.nexusmods.com/mods/29840/?
Then obviously my mod suggestions would be moot. As far as pure training if you don't want to look stuff up on the internet just start mixing potions. OR you could go around to the various alchemist shops in the land, buy some recipes (they are pretty cheap) and mix some of those potions. From there you should be able to figure out some ingredients that have common effects that can be mixed for new potions. Rinse, repeat.
Before you start your potion mixing you should go to the thief stone because alchemy is a thief skill. If you happen to see +alchemy items in the shops and can afford them, buy them. The potions you mix will be more valuable and thus you'll gain more experience.
The cost of the potion or poison made determines the amount of level gained.
Generally the potions and poisons that have mixed and off setting results cost more.
The potion I have found to make the most gold and level up the most is Salmon Roe+Histcarp+Salt pile.
You will need the Dawnguard dlc to get the Salmon Roe,but all the ingredients are easy to get.
Blue mountain flower and blue butter fly wing gives pretty good level ups also and is even easier to find if you collect the ingredients during your travels and make the potion when you get to a town.
But by far the best way to get your alchemy level up is to just walk everywhere and collect the ingredients as you go.
From there just keep making potions and experimenting to find the results you like.
This is a good section of the UESP. I do this very thing as She will then have the gold to buy the expensive potions.
As stated... Salmon Roe along with Chicken Egg and salt is good. I use chicken egg because they are relatively easy to get around Whiterun.
AS far as XP gains from eating an ingredient? I thought it did, albeit very very minute. However, it seems there is no documentation on it so I'm afraid I may have had it mixed up with XP gain from learning an enchantment. Either way, actually making potions is the way to go.
This right here^^
Plus, yes everytime you consume a new ingredient, you gain very very little XP to raise your alchemy.
I'm afraid that's not the case. Alchemy XP gain is solely based upon potion value.
However, if you can provide documentation or the equivalent in the form of some sort of test, then I will gladly accept. I thought it to increase XP as well. I was mistaken.
When I started playign the game I got the strategy guide. In it, it explains that your alchemy skill level raises the slightest bit everytime you eat a new ingredient to learn the first effect. Sorry I dont have the book in front of me.
Alchemy enchanted gear, create potions of Fortify Restoration. Drink potion, create more potions.
After getting the necessary ingredients, you should be able to get to 100 in like... 20 minutes?
But it does cheapen the experience. This is an RPG after all.
Good to see you as well.
Took a bit of time off. Magic the Gathering Pro Tour locked up a bunch of my time. Almost made the World Cup.
It can get to some pretty insane levels pretty quick. Have to watch it though, if you get it too high, it does the reverse and nerfs all of your bonuses...