There is a glitch (on PC, not sure about consoles) where if you buy things from a vendor, save, quit the game and restart, the vendor will be restocked. Tbh, I am surprised I have not seen this noted anywhere, on Youtube or on any forums, although I didn't look that hard.
With that in mind, I have managed to powerlevel both Alchemy and Speechcraft starting at level 1, with only a little money (500g is sufficient to start) and by visiting Whiterun and staying in the Alchemy shop while it is open.
So the process is:
1) Get 500g. If you are just starting out, loot everything that you can carry and sell it, including the gear you are wearing. You won't need it.
2) Go to Arcadia's Cauldron in Whiterun (the Alchemy shop).
3) Buy out of the following list to make these potions:
- Damage Magicka Regen
Blue Butterfly Wing, Blue Mountain Flower, Hanging Moss, Nightshade, Spider Egg, Spriggan Sap
- Fortify Health
Giants Toe, Wheat, Creep Cluster
- Invisibility
Chaurus Egg, Ice Wraith Teeth, Luna Moth Wing, Nirnroot, Vampire Dust
- Paralysis
Briar Heart, Canis Root, Imp Stool, Swamp Fungal Pod
- Slow
Deathbell, River Betty, Salt Pile
This is not a full list of components, just the list of things that the vendor actually sells that are relevant. Remember, any two of these items (under the same heading) will make the potion. The trick here is that these potions actually sell for more than the components cost.
The biggest money maker (and skill increaser) is Giants Toe + Wheat + Creep Cluster. Always get Giants Toes!
4) Go to the Alchemy Lab and make all the potions you can.
5) Save your game.
6) Quit the game (to Desktop!).
7) Restart the game. After restarting, the merchants stock will be refilled, although it refills randomly.
8) While still in the shop, sell your new potions to the merchant.
9) Go to step 3
Thats it! I tried this a few months ago and got 10000 gold and a high level within 2 hours. When you can, increase the Alchemy skills making your potions more powerful, and increase the speechcraft skills allowing you to sell for more and have cheaper component costs.
I have just now checked that they haven't nerfed it and it still works.
And also, to make the process of restarting the game easier, go into Game Folder\Data\Video and delete (or rename) the .bik file there. This will stop the Bethesda logo from appearing every time you restart.
