Every floor except the boss floors in the tower are the same. There are 4 rooms based on an element with two side rooms, and in the middle is a big square room. To get to the square room, you have to put a crystal/gem of the appropriate element in the middle of each of the 4 outside rooms. You get the crystals by killing the enemies in the 4 outside rooms.
The side rooms are where the chests are. You need a crystal to unlock the "doors." In those rooms will be a bunch of enemies, sometimes large enemies too. When you enter the room, stand still and wait for the room to load. Chests are directly opposite the door so just stand there until you can peek and see if a chest is in there. If so, run directly to the chest, open it and run out hopefully before you get killed.
When running these floors as a quest, it's easier to get the crystals by having more than one person helping in a room, because sometimes you don't get the right crystal that you need, and have to look for other enemies or wait for respawn.
If you are running alone with your story characters, at the logic stone near the room you're working on, set a homepoint. That way, if you die, you will spawn next to the room you're working on. Don't worry about dying, you'll just respawn again.
The Japanese PSN set rooms in towns by tower floor, so like create a room called "Vellgander1 chest" or "Vellgander1 Speed" if you want to run the actual quest.