Alchemy's a tough one to get into. You have to really dedicate yourself to being alchemy primary: make sure it stays as your highest skill, perk it up, collect ingredients constantly, make potions focusing on high yield buffs and poisons, and most importantly, actually USE the potions in every single battle. For an occasional user, potions you find and buy are generally better than ones you make, and if you training/perking it with tertiary priority, you'll have long since outgrown the potions/poisons by the time you make good ones. The same goes for Illusion, to a slightly lesser extent.
However, if you dedicate yourself to them, they are rewarding ways to experience a different style of gameplay.
Alchemy skill is based off the value of the potions you make. It's IMO one of the easiest to level up, on my current playthrough, I went from level 20 Alchemy to level 55 in a single potion mixing session. Theres a list on this site for Skyrims most valuable potions, and then theres another one on UESP wiki that will give you a list of most valuable potions. I actually do believe there might be a potion more valuable than Giant's Toe, Wheat and Creep Cluster, I think it uses Giant's Toe and Hanging moss, and a 3rd ingredient.
The easiest most valuable ingredients to find and level up alchemy, if you're not raiding every giant camp for giant's toes as well as stealing them (there's also 4 in Windhelm, Calixto's House of Curiousities and that mage in the castle)
I think it's Butterfly Wing, Hanging Moss and Blue Mountain flower, its one with the easiest ingredients to find in abundance and makes a very valuable potion. Blue Butterfly Wing and Blue Mountain Flower also makes a pretty valuable potion.
Basically when I'm out and about I never pass up Blue Mountain Flower and I never pass up Hanging moss, you can usually find those 2 ingredients in abundance (Blue Mountain Flower all over the place) and hanging moss in specific areas of Skyrim is always in great abundance, usually in the Reach, Solitude Crypt, and there's quite a bit at Eldergleam Sanctuary. I've collected so many butterfly wings at this point though that I don't really collect them anymore.
Also Dragon's Tonque and Fly Amanita makes a decent potion for leveling up Alchemy, and if you add Troll fat, even more so. Anything from a Hagraven, feathers or claws is good for making expensive potions but those ingredients tend to be hard to find. I think Taproot, Deathbell and Salt Pile makes a pricey potion.
Also your best poisons and potions are ones you craft yourself, assuming you're using the Falmer Helmet Glitch. When I get around to joining the Dark Brotherhood, something I've never done in 2 1/2 years of owning this game, I'm going to make some really powerful poisons. Never pass up Impstool, it's a key ingredient in all powerful poisons.
Really though if you want to know how to make a lot of use of the alchemy skill, use google, there are a lot of people who have experimented around with ingredients to figure out what makes a really powerful potion/poison and you can go from there.
as a side note, with my massive inventory of super valuable potions, I had to get the Trade and Barter Mod, as well as expensive investments mod so I wasn't so filthy rich for the whole game and money still had some kind of meaning. One of these days in my build, I'm gonna hit up the College of Winterhold and buy every spell at outrageous 20:1 Trade and Bart prices.
Also if you sell each potion individually, you'll level up speech really fast. Make sure you exit out of the barter dialogue and reenter after each time you level up. Speech is one of the fastest skills for me to level up with my method. Throw in a blessing of dibella and gift of charity too if you can, I'm still looking for that Amulet of Zenithar without hitting up the one in Riften.
http://skyrimforum.com/sf/threads/most-valuable-potions.4985/
http://www.uesp.net/wiki/Skyrim:Useful_Potions
http://gaming.stackexchange.com/questions/47339/how-high-can-poison-damage-go
^THis one is about making super powerful poisons.