You can also pay coin to have your skill leveled up for you. There are three Alchemy trainers in Skyrim, each of whom also sells ingredients:
Lami runs Thaumaturgist’s Hut in Morthal. She is an Adept level trainer.
Arcadia runs Arcadia’s Cauldron in Whiterun. She is an Expert level trainer.
Babette can be found in the Dark Brotherhood Sanctuary. She is a Master level trainer.
Above: If only reading books taught you stuff in real life...
There are five books to be found throughout Skyrim that enhance your Alchemy skill upon reading. Below you’ll find the locations for said books, though you only get the boost upon your first read.
A Game at Dinner
Song of the Alchemists
De Rerum Dirennis
Herbalist’s Guide to Skyrim
Mannimarco, King of Worms
There are a small number of quests that, upon completion, boost your Alchemy skill. They are as follows:
Discerning the Transmundane (+5 to Alchemy upon completion)
To begin this quest, seek out Septimus Signus in his outpost north of the College of Winterhold (pictured above). The quest is lengthy, but at the end, choose the ‘path of the thief’ for your stat boost.
Lami’s Misc. Quest (+1 to Alchemy upon completion)
Lami (she runs Thaumaturgist’s Hut in Morthal) is looking for a book called “Song of the Alchemists.” See the previous section for this book’s possible locations. Giving her the book, boost your stat, but don’t forget to read the book first for an additional alchemy boost!
Arcadia’s Misc. Quest (+1 to Alchemy upon completion)
Arcadia (she runs Arcadia’s Cauldron in Whiterun) is looking for some frost salts, which are dropped by Frost Antronachs.
Frida’s Misc Quest (+1 to Alchemy upon completion)
Frida (found at The Mortar and Pestle in Dawnstar) is looking for the Ring of Pure Mixtures. You can find it inside the Forsaken Cave (pictured above).
Inge Six Fingers’ Misc Quest (+1 to all stealth skills, including Alchemy)
Inge, who can be found at the Bards College, is looking for Finn’s Lute. You can accept the quest to find it from her directly, or find the lute first and present it to her. The Lute can be found in a chest inside Stony Creek Cave (location pictured above).