The simplest ways are pinking the edges (need to purchase pinking shears, only works on tightly wove fabrics), and zig-zagging the edges, which is kind of like a poor woman's serger.
There are other types of seam finishes I've heard of but never tried, such as a Hong Kong Finish,which I've simply never had occasion to use, but h ope to try sometime in the future.
One of the coolest finishes I've learned is the ones you'll see on most pairs of jeans, called a Flat-Felled seam (tutorial).
Also, I think it is a very nice touch to learn hand blind stitching for hems, sleeves, and neck facings, but I'll get into that later
It takes quite a lot of time to do the proper pressing, extra seam finishing, and hand stitching on a garment, but your finished product will show all the extra love, work, an attention you put into it!
Happy sewing everyone!