I need help with the Full Moon quest..how do you find the Abandonment Hut? Every Friday around 18:00 to 23:00 I have to meet Bernard to find Caleb with him.
That sounds like your normal Friday hunting trip with Bernard. That's part of the normal relationship route with Bernard which is required before you can work on the Caleb quest. Meet Bernard every chance you get at the cottage in the woods Friday nights, at the tavern after you talk to him about a night out, and regularly at the hot springs every Saturday and Sunday evening (let him pay or he'll be offended and like you less).
Repeat those steps and you'll eventually get different dialog in them and a notice that he loves you more (as in, you've gained a heart).