From my experience, there are several ways to do this. The most simple way is to have your character's hands/arms a separate sprite and then rotate them to look at the mouse. Then you can spawn bullets/arrows/whatever from a child node of the character's hands/arms, and then move the projectile according to the direction it is facing.
While the code is for Godot 2.0, I would take a look at the LookAt.gd file from the Godot 2.0 GitHub demo repository, as it shows a few different ways to rotate a sprite to look at the mouse. You may need to convert a few things if you are using Godot 3.0, but it shouldn't be too hard to do. Let me know if you have any issues and I'll see what I can do!
Here is a Godot QA answer that has a example project that appears to aim at the mouse. I have no idea which version of Godot it is for, though looking at the posting date it is probably Godot 2.0 as well.
There is also a quick, slightly over 1 minute Youtube video showing how to make a Cannon head aim at the mouse with Godot 3.0.
For moving the node in the direction it is facing, I posted some example code in this topic awhile ago. I think the code should still work.
Hopefully this helps! :smile: