You can access these sidequests after completing The Power of the Kings during Chapter 2. Talk to Cindy at Hammerhead for a series of sidequests to upgrade the Regalia.
The Ever Regal Regalia
Unlock Condition: Reach Chapter 2 and complete The Power of Kings
Reward: 150 EXP, Aero Wax component for the Regalia (improves fuel efficiency)
Talk to Cindy. Then head for the tunnel you can access form the south of the Prairie Outpost to retrieve the Aero Wax. You will encounter Goblins along the way. Once you’ve retrieved the item, return to Cindy. You can install the Aero from Cindy’s customize menu when you’ve completed the quest.
The Ever Elegant Regalia
Unlock Condition: Complete The Ever Regal Regalia
Reward: 500 EXP, Hammerhead Decal, Racing Stripes I Decal, Basic Color Samples
Talk to Cindy. She’ll then ask you to head to Balouve Mines and retrieve the Redstone Shard at the dungeon’s entrance right next to the elevator. Return to Cindy after you’ve finished.
The Ever Gleaming Regalia
Unlock Condition: Complete The Ever Elegant Regalia
Reward: 600 EXP, Fiberglass Coating component (keeps the Regalia clean)
Talk to Cindy, then go to Alstor Slough (a short way to the southwest of Coernix Station – Alstor, once you are in Chapter 3 or futher). The Fiberglass coating is found near an old car next to a small house by the lake. Retrieve it and return to Cindy.
The Ever Valorous Regalia
Unlock Condition: Complete The Ever Gleaming Regalia
Reward: 1,000 EXP, Supercharger component (increases the Regalia’s top speed)
Go to the specified area in the west part of the Vesperpool (only accessible from Chapter 7 or further). Get the component lying on the dirt road near a car wreck. Return to Cindy afterwards.
The Ever Illustrious Regalia
Unlock Condition: Complete The Ever Valorous Regalia
Reward: 3,000 EXP, Enhanced Headlights component (repels Daemons when you drive at night)
The next location is in the depths of the Crestholm Channels. Be sure that your party is at sufficiently high levels to explore the area. The item to be retrieved is in the room to the north of the second control panel. After you’ve obtained the item, return to Cindy.
Into Unknown Frontiers (Unlock the flying Regalia Type-F)
Unlock Condition: Complete “The Ever Illustrious Regalia” and “Formouth Garrison”, then finish the main story.
Reward: Regalia Type-F
After clearing Formouth Garrison, explore the base and look for a Strange Engine collectible. If you haven’t done so already, return to Aracheole Stronghold and get the Warped Wings. After collecting all three unusual parts (the Unstable Stabilizer is automatically obtained when you clear Fort Vaullerey in Chapter 6), talk to Cindy. You will get the Regalia Type-F.