Which alchemy perks actually help you level alchemy faster? I was told the first 3 perks do, but have had no confirmation (or if it actually even works)
Making stronger potions and poisons (e.g. a higher gold value) gives more experience. Thus, all perks that makes your potions stronger also helps you gain more experience per potion. These perks are Alchemist, Physician, Poisoner and Benefactor which all boosts the strength of your potions. The Alchemist perk is probably the best, since it boosts the effect of both potions and poisons whilst the others only boost certain types of potions.
Physician makes potions that restore Health, Magicka or Stamina better, Poisoner makes poisons more powerful and I believe that Benefactor makes all potions that enhances your stats better. However, Alchemist is needed to get the Physician perk and Physician is needed to get the Benefactor and the Poisoner perk.