Canis Root + Swamp Fungal Pods = Paralyze Potion
Salt Pile + Deathbell = Slow Potion
Boom. Useful and valuable potions that level you up pretty quick. Though, I leveled mine pretty quick using a mixture of Blue Butterfly Wing, Blue Mountain Flower, and whatever ingredient I had on me that boosted the value even more. I was making potions worth over 1k from the blue ingredients alone and the benefactor perk.
For me it kills immersion to make recipes you heard about outside of the game. Buy recipes from alchemists, and change your mindset. You're not making useless potions, you're using your craft to make money, and experimenting with different ingredients to make yourself a better potion maker.
If you've got Hearthfire, just get some Salmon Roe + Garlic and Nordic Barnacle. Almost every potion you make will increase your skill.
What I like to do is find a handful of recipes useful for that playthrough and use them exclusively. That way I don't need to memorise too much and only harvest what I need.
Whenever I use Alchemy I find it helps to place perks into it early on, to get potions that can increase my damage output as well as healing potions. If you play a race that starts with Alchemy, and put the first 3 perk points you unlock into that skill, you can make potions that heal around 50 health. An abundance of these makes up for having less perks in your combat skills.
You could play a sort of Hunter character who uses archery to take out monsters with rare ingredients (Trolls, Giants, Spriggans etc.) and makes potions that boost his skill. I find taking alchemy as a pure warrior isn't as fun. If it's you first time taking Alchemy I recommend a Light Armoured Ranger type character.
Look for recipes on the web, e.g here on the blog. If you are pissed off of leveling the skill, level it with trainers for free. Simply pickpocket the gold back. You have 2 ways to do it:
-Without any pickpocket skill: Paralyse the target with spell or with Ice form. As this does not deal any direct damage, they won't turn hostile. When they are getting back their conciousness, you can pickpocket them with maximum success rate. (Credit: Ponty)
-Pickpocket skill. I, myself, have done it the following way: I had the very base thieves guild gear, 3 pts in light fingers, 1 night thief and 1 cutpurse (pickpocketing gold 50% easier) Pickpocket is leveled very easily and gives a lot of XP to the character level.
This way you can level alchemy without spending a lot of time leveling it, and it's free. (I recommend the same methode to all crafting skills as they are very boring to power level and it takes much time).
I hope this can help