online

Rule of thumb. Always keep a sharp line between what is visual and what is behind the scenes. It's understandable that you want to keep both of them as in sync as possible, but critical information such as who did what damage is more important to keep in sync then assuring that it appeared like you hit the person.
If you put everything into making it look like you hit the person, then rely on local hits to do any damage, theres a sharp chance of you breaking sync anyway, because it's still possible for visual lag, especially with radius attacks that do different damages from different locations.

I made a few online action games and I can say that you MUST use UDP for sending movement. It is unsecured and there is no guarantee you'll receive it, SO you have to send data in a continuous loop.
Other things like HP, shooting, etc. need to be sent using TCP without any loop.
But we're a little off-topic now. There are some examples of using MOOGame on this community. You can check out my example showing how to make a simple action game using it (also available on this community).

What would be the best way for me to make my game work over a LAN? Is there somthing (like mooclick) that would do that and is it easy?