So I will be using PlayStation 3 Layout to make sure stuff is unambiguous, as e.g. between Nintendo and Xbox the buttons are similar named but at different spots which is annoying to say the least.

These should work with Steam Input off (depending on the controller) or with Steam Input on for any controller that supports it.

Events:
Cross - Advance
Circle - Hide Dialog
Triangle - Backlog
Square - Auto
Start or Select - Skip the entire event
R1 (hold) - Fast Forward
D-Pad/L-Stick Left/Right - Text Speed

Dungeon:
Cross (tap) - open Items
Cross (hold) - Run

Circle - Attack
Square - Aim (can be combined with force diagonal)

Triangle (hold) - shortcut menu (use L-Stick or D-Pad to select specific options)

R1 (hold) - Force diagonal (can be combined with Aiming)
R2 (hold) - Dungeon Backlog
L2 (hold) - Opens a currently unknown menu
Start or Select - (Persistent window with Dungeon Backlog, Dungeon Info, Dungeon Settings and Options)

Cross+Square - Yuragi Dash (Auto-Dash until the next Intersection/room/etc)

Square (in Items) - Sort (also combines throwing items)

L1 - Throw equipped throwing item

Here a List of the Keyboard Buttons and the Controller Buttons:

Q - L1
E - R1
WASD/Arrowkeys - D-Pad and/or L-Stick
Y - Cross
N - Circle
R - Triangle
F - Square
B - R3
Z - L2
C - R2
I/K - Right Stick up/Down
(likely J/L R-Stick Left/Right but so far not used)

Tab/Esc - Start/Select