It's important to secure the ends of a sewn seam, so your piece doesn't fall apart.
In the case of a sewn seam, it's best to backstitch at the ends. On your
machine, there should be a lever or button you can operate that will cause
the machine to feed your cloth backwards temporarily.
Make the seam by putting the cloth under the foot, with the edge at the back of the foot - this should give you enough cloth to let you sew about nine or ten stitches backwards to the end, before releasing the backwards-feed and sewing normally until you reach the other end, when you engage backwards feed again for nine or ten stitches.
Simply trim the ends of the threads - the sewing will be close enough to the backwards sewing that it won't work loose (and if it's zig-zag, it'll most likely cross over itself to make an outright knot). You can use a needle to pull the ends of the thread through to the same side and reef knot them if you're paranoid.
This backstitching will make the ends of the seam stronger than they would otherwise be, as well as securing the thread ends.
You shouldn't have to secure a serged seam in principle - your tension should exactly match the stretch of your fabric, so when the cloth is fully stretched, the threads should just be taut.
Myself, I find this is tricky to arrange, so I tend always to tie a simple overhand knot in the chains, using a needle through the loop's knot held against the seam so that while I'm pulling the chain's end, the knot tightens right against the seam. You can get special glue for this as well.
Serged corners and curves
For sharp corners and curves, although pinning is a must, you shouldn't let pins go through the serger. I've found it helped to pull the edge into the path of the serger's needles by pushing a threaded needle through both top and bottom piece of fabric, and pulling both ends of the thread. Once the point has gone through the serger, you can easily draw the pulling thread out.