See Controls for a more complete list, here are the basics:

Move:
Press the [Left] and [Right] cursor keys to walk.
Press [Enter] to jump forward.
Press [Backspace] once to jump upwards.
Press [Backspace] twice to jump backwards.

Aim:
Press the [Up] and [Down] cursor keys.

Aim precisely:
Hold down the left shift key while aiming

Shoot:
Press [Space].

Switch weapon:
Press [F1]-[F10], or use the weapons menu, accessible via right mouse button click

Switch between hedgehogs:
See “How do I switch between hedgehogs?”.

Turn around without walking:
Hold down [left shift] and press the [left] or [right] cursor key.

Adjust sound volume:
[0] key is for volume up, and [9] key is for volume down

Set timer of e.g. grenade:
Press [1]-[5] keys

Set weapon bounciness:
Hold down left shift and press [1]-[5]

Control the direction of air attack:
Press [Left] or [Right] cursor keys

Construct different girders:
Press [Left] or [Right] cursor keys when using construction tool

Note that you can configure most key bindings in Hedgewars, but you cannot change the key combinations directly.