What is the fastest way to level up my alchemy skill?

I'm fairly sure discovering new effects gives you the biggest boost to your alchemy skill, but throwing together your own potions is gonna be effective, of course.

My strategy has been to go out and collect too many ingredients and then head to an alchemy table and just start throwing stuff together. Generally, I try to put ingredients that I know a lot of effects with ingredients that I know zero or one effects. This way, I have a better chance of matching something.

Eating ingredients will uncover the first effect.

Oh, and you could always train from a specialist and then kill them and recollect your gold. But that's limited to fives times per level.

This wiki is a great source: http://www.uesp.net/wiki/Skyrim:Alchemy

My Potion recipe finder generates all possible 2 and 3 ingredient recipes and lets you filter by effect and purity. You can remove items you don't have and it adjust the possibilities. It also sorts by price so the post expensive ones are first.

The amount of exp you receive is directly related to the amount of gold that the potion is worth.

Potions are worth more if they have more effects on them, and the effects have a larger magnitude.

For this reason, even creating a potion that has 1 positive effect and two negative effects will be worth more than a potion with just one positive effect.