Let's talk seam finishing. Why do we finish seams? It's an important step in sewing--unfinished seams lead to fraying which can weaken your hard work. There are several ways to finish seams so that your garment is comfortable to wear, and will last a long time--and only one of these options includes buying a serger!
If you're new to garment sewing, start with the tools you have--pinking shears or your own machine. After sewing your seam, use your pinking shears cut the seam allowance by about half. The zigzag cut will keep your fabric from fraying.
Likewise, you can use the zigzag stitch on your sewing machine. After sewing the seam, press the seam open, and zigzag stitch the length of each seam allowance. If your machine has an "overlocker" stitch feature (and many do), you can also use this.
You can also use a special overlocker/side cutter foot that mimics a serger. The difference between using the overlocker or zigzag stitch and doing the same with the overlocker foot, is that the foot actually has an attached blade to cut the excess fabric, thereby eliminating the seam bulk.
Here's a short video demonstrating our overlocker/side cutter foot.