I will be doing a more in depth post on alchemy, but for now here's a tidbit:
To level up in alchemy the fastest start buying all ingredients (except daedra hearts and void salts, EEEK expensive!) and make as many potions as you can, use my cheat sheet for help with getting this done faster. Keep selling the potions that you don't need and then when you have 5,000 coins, go to Falkreath and talk to the jarl. Out of the three houses, this is the easiest one to get (I like to get all three but I always start with this one first). Once you do whatever quests required to purchase the land (usually killing off some bandits for the jarl), by the land for sale, build your small house and then the garden. Plant two different ingredients from the following list in all the available spots:
Blisterwort
Blue Mountain Flower
Imp Stool
Swamp Fungal Pod
Wheat
(I suggest wheat, imp stool or blue mountain flower since they are easiest to get)
The combination of any of these two will give you health potions which you will be so glad you have. You'll get about 3-4 ingredients each time you harvest which will make more Restore Health potions than you'll know what to do with. This also helps increase your alchemy.
You could also do this with ingredients for Restore Magicka or Restore Stamina potions if your character will require them on a regular basis, but I suggest starting off with health potions.
Your health potions will get to the point where they will be stronger than potions you'd by from an alchemist, woohoo!
EDIT NOTE: If you do blue mountain flower and wheat, it will give you a potion that will both increase your health for a certain amount of time AND restore a certain amount of health.
