A serger is great for sewing with knit fabric, which has stretch. But you can achieve just as fine results with a regular sewing machine. Just consider these things:
Using the right needle
The most important thing is to switch out your regular needle on the machine for a ballpoint/jersey or stretch needle. They have a rounded tip which won’t ruin the fabric. The same goes for your twin needle, if you’re using one. That one should also be ballpoint/jersey or stretch.
Use zigzag stitch
This is really important, especially for horizontal seams where stretch is needed (e.g. the neckline). A seam that doesn’t require much stretch, e.g. vertical seams such as the side seams, can be sewn with a narrow zigzag stitch.
Don’t pull the fabric
Try not to pull the fabric as you’re sewing. That generally results in a “bubbly” seam. Neither should you let the fabric hang down from the table. Lift it up so that it can be easily fed into the machine.
Always press a seam after you’ve sewn it. Don’t wait until the entire garment is finished. Pressing is often the thing that takes away that “homemade” look and gives the garment a professional feel. When you’re pressing, the thread “melts” into the fabric.
Change presser foot
If none of the above helps you could try switching to a walking foot. It keeps the fabric from stretching as you sew.