I found that I was able to get Alchemy before Mining and Refining; while this seems like it would not be a huge problem, one of the quests I received under Alchemy required me to make Fertilidust, which requires chemicals that need Shiny.... which as far as I could find, does not exist in stores. The only way to get Shiny was to mine for it and then refine the results, which, of course, I could not do, nor was I told of this until I got some help from the other players (thanks, other players!).
Personally, I would not mind seeing the Skill Tree altered so that a player needed Mining and Refining as prerequisites for Alchemy. Alternately, I would not mind seeing Shiny available through Street Spirits--if I somehow failed to see this, of course, I apologize for being partially blind.