I only have limited applique experience but this is what I know - when using (non-stretchy) cotton, you can use an iron-on interface like vliesofix to adhere your top layer to the bottom layer, the interface helps stop the seams from fraying. You can then go over the edges with either a blanket stitch (by hand or on your machine) or any other stitch you like like a satin stitch. A good sewing machine makes the job a billion times easier (could never do it on my old machine, can do a half-tidy job on the new one).

In the example you gave of the square, if that's the only thing you're doing, I would iron a quarter-inch seam all the way around BUT cut a small triangle off each corner before you do, so you don't get any bulk, then pin and sew it on that way. The other thing you can think of is getting a rolled-hem foot, which turns the fabric under and allows you do a tiny little neat seam, probably not needed for this project but handy if you're ever finishing seams on open things like scarves.

Or the other way (that quilters use) is to use what's called "needle-turn" where you sew the hem under by hand as you go.