Here's a handy little trick for tucking in the serger threads on your projects. Take a large needle with a large eye and slide it into the seam, about an inch or so. Thread the serged tail through the needle and pull it through. If you want, you can repeat to really lock it in. I learned this from one of my sewing friends, and it's been the best solution!
Another way to finish threads is to tie them in a knot, although this works best where they don't show.