Alchemy is fairly simple to get to 110 but there isn't an updated guide so here's one to use to take advantage of the January crafting campaign!
As per the other guides:
* Always use Kitron Macarons
* Synthing pre-70 is a nearly guaranteed skillup so ride cheap synths as far as they'll go
* You do not need any subcrafts for this guide, but you will want to level Bonecraft and Clothcraft at a minimum once you are 110.
Nearly every recipe I'll use comes from ingredients you can get from shops, but there are a few things you should consider stockpiling to make the later levels a bit easier:
* Dragon Hearts
* Revival Tree Roots
* Fresh Orc Livers
* Philosopher's Stones
* 40k Guild Points, you will need Alchemic Purification and Anima Synthesis to be able to craft all of the materials for your final rank-up item
The main crystals you'll need are Ice, Water, and Light. Farm clusters in Sky.
All recipes will start at 5 levels below cap and end at 11 above cap. Pick and choose when you want to switch based on how much gil you want to sink and how much you feel like changing areas to get new ingredients.
0-17: Tsurara (Distilled Water x2 + Rock Salt + Ice Crystal) : Buy in Adoulin or wherever, really
13-29: Poison Potion (Mercury + Poison Dust + Water Crystal) : Buy from Alchemy guild
24-40: Yellow Textile Dye (Turmeric + Water Crystal) : Buy from Tenshodo merchants in Norg/Low. Jeuno/Nashmau
29-45: Artificial Lens (Glass Fiber x2 + Fire Crystal) : Buy from Alchemy Guild
35-51: Firesand (Bomb Ash x2 + Sulfur + Earth Crystal) : Buy Sulfur from the Gustaberg vendor or just use the Alchemy Kit 40
36-52: Prism Powder (Artificial Lens + Glass Fiber x2 + Light Crystal) : Do these if you made Artificial Lenses and you don't want to just NPC them
46-62: Holy Water (Distilled Water + Light Crystal) : You may want to start these even earlier than 46 with support and deal with the break rate - stockpile these for future synths and take this to the cap
62-79: Venom Potion (Venom Dust + Mercury + Water Crystal) : Buy Venom Dust from the goblin in Lower Jeuno and power this through
74-85: Paralyze Potion (Paralysis Dust + Mercury + Water Crystal) : Use this if you want to save sparks for later, buy Dust from Jeuno goblin
78-94: Holy Lance (Mythril Lance + Holy Water x2 + Light Crystal) : Buy Lance from Sparks NPC and NPC results
79-95: Dragon Blood (Dragon Heart + Lightning Crystal) : Use this if you stockpiled Hearts
86-102: Mamushito (Animal Glue + Paralyze Potion + Shinobi-Gatana + Water Crystal) : Use stockpiled Paralyze Potions, buy Glue from guild and katanas from Sparks NPC and NPC results
92-108: Ice Shield (Rock Salt + Distilled Water + Diamond Shield + Ice Crystal) : What an amazing synth - can start as early as the mid-80s if you have the skillup items. Buy shields from Sparks NPC and NPC results
108-109: Panacea (Mercury + Philosopher's Stone + Rock Salt + Sulfur + Light Crystal) : Only do this with gear or support so you have a chance to T1 it
108-110: Cantarella (Distilled Water + Fresh Orc Liver + Mercury + Paralysis Dust + Venom Dust + Dark Crystal) : Cap with this if you stockpiled Livers
108-110: Ephedra Ring (Holy Water x2 + Hallowed Water + Mythril Ring + Light Crystal) : Use this if you made a ton of Anima and still have Holy Water stock after making your rank-up ring
108-110: Azure Cermet (Cermet Chunk + Panacea + Azure Leaf + Fire Crystal) : Azure Leaf is ~1.1 million per item from the guild, used for Omen upgrades. May want 116+ effective skill before trying since it should be HQable.
You can also get to 110 easily with Sun Water if you have a well-leveled Mog Garden and the ability to buy Cactuar Roots and Beastman Blood with the proper Mementos.
At 107+3 you are Tier 1 on every Cursed -1 synth possible and 110+12 will get you to T2 on Voodoo Greaves and Jinxed Nails synths that use the respective -1 Cursed/Hexed piece, as well as 121 effective skill to break T1 on every Hexed -1.