This build gives enough AP early to be able to clear waves effectively, but we are not necessarily playing for damage (You may do some decent damage early and mid-game especially if you are ahead, but you will absolutely fall off). Transcendence and Jack of all Trades allow for effective Surround Sound spam with tons of ability haste (Seraphine's best stat).

We play very intentionally to obtain our 5 and 10 unique stats for Jack of all Trades. Thus, first back we definitely want to get one of the early spread of items that provides that as soon as we can which we can do with as little as 750 gold, but ideally we back closer to 1000g, but this can be hard since Sera runs out of mana very quickly early. If we can get 1500g then lane is set up very well for success since Lost Chapter is a critical powerspike we want to delay as little as possible.

Mercury's Treads are crucial to obtaining 10 stacks for the additional 25 AP, because they are the only boots that give three stats. Mercs are kind of expensive, but I don't feel like the setback is that bad especially since Ionians are kind of meh this patch with the 10 AH, Sorcerer's Shoes are probably possible to run, but probably a weaker utility since enchanter doesn't really play for damage anyway. Also you would have to find another stat relevant to Seraphine somehow in your build which would probably have to be % Magic Pen, Armor, or MR, none of which can be found in any enchanter items anymore.

You play for your mage capabilities early game with Seraph's Embrace, but we go Cosmic Drive after for the sake of ability haste mostly, but also because it accomplishes a lot for the stacks of Jack of all Trades while being pretty helpful for both the mage and enchanter elements of your kit. Playing for mage early and enchanter late is strong with competant teammates since your mage strength falls off compared to other mages, while your enchanter capabilities only matter after Surround Sound has some point investment from level (which Triple Tonic accelerates).

Jack of all Trades when fully stacked gives 25AP and 10AH, which is ~1000g worth of stats and equates to a free Fiendish Codex. After Doran's Ring, Seraph's Embrace, and Mercury's Treads we have 7 unique stats (Health, AP, Mana, MS, AH, MR, and Tenacity) getting a Forbidden Idol gives us 2 more (Mana Regn, Heal/Shield Power) and Aether Wisp gives us our 10th and final stat (%MS), we can also sell our Doran's Ring to complete the Kindlegem of Cosmic Drive if needed, but otherwise we shouldn't sell Doran's until we have Cosmic because then we wont have health anymore, which is one of the 10 we need.

Alternatively you could just combine the Aether Wisp and Forbidden Idol into an Ardent Censer, but only if your team auto attacks quickly and would benefit from Ardent. Ardent also doesn't give any ability haste, but this is probably fine since you are fully stacking JOAT off your starter item, 2 legendaries, and a pair of mercs which is way earlier than Jack of all Trades is typically intended to be completed.

Jack of all Trades is kind of hard to fit as a mage without botching your build, but its a pretty great rune on mages if you can manage it. Merc's are the key item that allow this build, while also being a much smaller drawback than most other paths to 10 unique stats which usually involve bigger commitments to legendary items that don't benefit enchanter apc very much. (e.g. Cryptbloom, Banshee's Veil, Zhonya's Hourglass, Spirit Visage) If Merc's are SUPER not worth it in a given game then probably just run the Legend: Haste variation, but Merc's allow this stacking to happen while running actually good enchanter legendaries that we would want to run anyway, while further contributing to the survivability of your Seraph's passive shield, Cosmic's health and mobility, and AOE shielding bolstered by enchanter legendaries.