You need to have the righting line attached around the base of the mast before you go sailing. How you store the rest of it is up to you. My G-Cat has a 3-sided rear-opening pocket on the trampoline just aft of the mast. It is just large enough to stuff a floatation cusion into, and I stuff the rest of my righting line under the cusion to keep it in place. My righting line is a piece of dock-line from Wal-Mart I think. It came with a loop in one end for use on mooring cleats. I put the loop end around the base of the mast.

Find the book "Catamaran Sailing: From Start to Finish" by Phil Berman. It is the book to own when you're starting out in this sport/hobby.