Back stitch is going to be your best bet for keeping the garment together in general. THere are some hem stitches that may work depending on what you're doing.
You didn't mention if you knew the backstitch so I'm going to go ahead and give you a run down of it. If you already know, ignore the next paragraph.
First do your regular in and out stitching to get the pieces together (running stitch) which is the most common hand sewing form. Back stitch is where after you make the starting point of your stitching, you (considering the stitch measurements are in millimeters) 1) go forward and insert your needle 2 millimeters. 2) go backward almost within the same point as where you started and insert you needle. 3) move forward about 4 millimeters and insert needle. 4) repeat the steps by going back 2 millimeters again.
Also, double your thread by tying the two ends of the thread together.
As for edgestitching, a good replacement would be the overcast stitch. You just sew loops over the edge of the fabric. Depending on the width between the edge and the stay-stitch, you can probably overcast both edges together to give it a finished, serged look.
Here's a pic of the stitches and more from the internet to help. If you need more, I have a book on this stuff that might help, but I just don't have it on me at the moment.
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