Prestige and alters are related in the experience description and separated on the mechanics board. Read both before you spend a Mirror Fragment.
What prestige does
The prestige card says it gives the prestige currency and does not reset perks, characters, or emotes. Quest text and codes call the currency Prestige Carats. The prestige card and the daily-shop card spell it Prestige Carrots. Treat those as the same spendable until the game shows two wallets.
The experience page says that with the right Uma equipped, prestiging might reveal a hidden alternate form. It does not name the Uma or the cost.
The documented alter path
The Alter Umas card names two alterable characters right now: Tokai Teio and Haru Urara. Talk to Director Yayoi Akikawa in the lobby for the requirement and the quest tracker. When the quest is complete, you are prompted to spend Mirror Fragments.
V0.11 increased alter requirements and slightly nerfed Alter Haru's skill. The new requirement numbers are not printed. Do not reuse an older checklist from memory.
HARIKITTEIKOU! is the only code recorded to grant a Mirror Fragment, and even that reward line disagrees about Spins versus Carats. Other Mirror Fragment sources are not listed.
What the alter kits add
Alter Haru has one dash, a one-time skill, and the ultimate ONE VICTORY at 1.48x speed for 12 seconds with 70% stamina restored. Passing during the skill is the burst condition. Passing during the ultimate debuffs the other racer.
Alter Tokai Teio has two dashes and Certain Victory at 1.55x speed for 13.4 seconds with 65% stamina restored. The skill line adds a short burst and reduced stamina consumption on top of speed and recovery.
Base Haru and base Teio stay in the roster. Alter is not a replacement record.
Prestige shop
The public-beta notes put prestige perks in the prestige shop for Prestige Carats. Prestige Spins, also called Lucky Spins in that note, cost 5 Prestige Carats each and are reached by scrolling that shop. The Uma spin tab has a switch that spends Prestige Spins. Daily and hourly shops can also take the prestige currency.
All-time quests pay extra Carats at 1, 3, 5, and 10 prestiges. Those payouts are on shops and quests.