The goal is always to sing from your diaphragm. This is a system of muscles at your lower ribs that acts as an inhalation device. Singing from the diaphragm is the only way to sing correctly, without damaging your throat, and powerfully. Get familiar with your diaphragm by finding a quiet, comfortable place to sit. Slowly, by breathing through your nose, bring air deep into your lungs. Fill your lungs completely, then slowly release the air through your mouth. Do this several times. You'll feel your diaphragm expand with your lungs. This well help relax you before you begin exercising and singing.
A good exercise that helps you use your diaphragm to control breathing while also helping you with enunciation is to breathe in deeply and sing a vowel. Let your breath out slowly behind the vowel you're singing. As you near the end of each vowel, allow your diaphragm to push air out a little faster and change the pitch of the note you're singing. Try bringing it up a half or whole step. It will be easy to notice how much stronger the note your singing comes out stronger than when you try to sing it from your throat, which is a common mistake of many amateur singers. This exercise, because it helps with two elements of singing, is a good one to practice regularly.
Lie flat on your back with your hands on your waist. Point your fingers toward your navel, and breath in, filling your stomach with air from the bottom to the top. You will probably need to make a conscious effort to do this if you are just starting out. Breathe through your nose, and if need be, bring your stomach inward just slightly, forcing the air to start at the bottom. Don't take as deep a breath as you can; you don't want so much air in your lungs that you can't do anything with it. The object here is to get accustomed to the feeling of air in your stomach rather than your chest. Hold the air for a five count and then let it drift slowly from your mouth. Your shoulders should not rise when you breathe with your diaphragm. If they do, you're still using your chest to breathe.
Scales are still one of the most effective ways to exercise your breathing techniques. If you can sing scales by ear, practice singing them as you work on bringing air into your diaphragm and releasing slowly or more quickly, depending upon the power with which you want the notes to sound. If you aren't quite to the point of singing scales by ear, you can purchase a tape of scales from a local music store and try singing along until you can match the pitch. You'll be surprised how much easier it is to sing in tune when you are breathing properly.