What are the best / most recommended trios of crafting professions for each class?
3) Current trio: Foraging, Apothecary, Artificing. - "I like to pick flowers and smash them into cocktails, then blame them when I break my accessories."
"What should I be doing craft-trio-wise with these mage builds?"
First, your calling matters not! (The fact you're a Mage has no bearing on which craft skills to select.)
My mage is an Apothecary and I know that once you reach Savant level the only gathering skill you need is Foraging. (Forage Sarleaf, lots and lots of Sarleaf!) Apothecary is easy to zoom up to 450 (from 375) if you have about 7 stacks of Sarleaf Extract. I'm not sure if another gathering skill is needed below 375 because I purchased a level 60 character pack and it came with 3 tokens to auto-grandmaster a crafting skill.
As for Artificing, you REALLY want Mining to go along with that. I'm not sure if you can buy the gemstones you'll need from the AH (Auction House) for reasonable prices. Maybe you have tons of plat and don't mind paying the AH costs, I don't know. If you ask a Trainer -- find the Artificing trainer in your city and 'Ask about Artificing' -- they'll tell you which gathering skills work best with that crafting skill. Good news is you can buy a 4th (and 5th!) crafting skill with Credits from the Rift Store.
Outfitter and Runecrafting don't require a companion gathering skill. Outfitter creates cloth and leather armors. Leather armor requires Butchering to harvest Hides, which are crafted into various Leathers used in Outfitter recipes. However, cloth drops from dead mobs (undead-type mobs in particular) so you can raise Outfitter skill by crafting only Cloth armor. And Runecrafting requires special fragments and crystals, which you can get by "Runebreaking" the cloth armor you make. Runecrafting gets crazy expensive once you reach Grand Master level, so be warned. Since you're a mage, you wear Cloth armor so you can wear your own armor. Also, Outfitter makes a lot of things that benefit Mages (Trinkets, Seals, Augments etc etc.) Regardless, know that the best armor in the game drops from mobs in Dungeon and Raid rifts, so you'll have zero demand for any armor you make.
I'm not a Rift crafting expert, but I think all other Crafting skills (other than Fishing/Survivability) require one of the gathering skills.