This guide has the gates, red and blue (or only one of them, if there was only one in the map) to locations in quest mode. Hope it helps!
Contributions to unlisted locations welcome, as well as comments and suggestions.
Special thanks to: CuzItsSweet and darkding for helping me find some of the gate locations.
Quest mode tips:
Having dual spiked wonder bit is invaluable. It'll protect from virtually all melee attacks and 1-hit or 2-hit KO almost all mobs after getting a few WB Boosts until area 60. Even after, having it will prevent melee-attacking mobs from ganging you.
Buy Packet mate from the UC shop for 2000 UC. Equip her and set her support mode to Gate Radar.
If necessary, keep a supply of Repair 50% on one item slot and preferably Speed Up on the other. Sometimes running away will be vital to survival.
Winberrl Arms are useful for catching "Dash" keyholding bots.
Invincible is by far the most useful item in Arcantus Quest mode and it should be held at all times when found. If you already have 4, use one and grab the one on the ground. If you find a gate radar, but have full items, hit Tab to drop one of the Invincible items, use the radar, then pick it back up.
Arcantus:
Area 4
Area 8
Area 11
Area 12
Area 14
Area 17
Area 20
Area 24
Area 26
Area 29
Area 32
Area 35
Area 36
Area 38
Area 42
Area 44
Area 47
Area 48
Area 50
Area 53
Area 56
Area 58
Area 60
Area 62
Area 71
Area 72
Area 74
Area 77
Area 80
Area 83
Area 84