This sounds simple, but when you're up on stage, it's easy forget to inhale all the way. Breathing deeply means having the proper posture and support to let as much air as possible into your body. When you breathe, your diaphragm should expand to let air in.
You can strengthen your diaphragm with this exercise: Stand facing a wall. Place a medium-sized book between your stomach and the wall, parallel to the ground. Breathe in and out. If you're diaphragm's working properly, you should be able to do "pushups" with your stomach.
You may have taken a good deep breath but it will be lost if it comes out too soon. You have to breathe evenly in and out to sustain longer passages. To practice this, breathe without singing. Take a deep breath and then try to let it out as slowly and with as little noise as you can. Don't let the muscles of the abdomen and diaphragm relax as you exhale -- they should stay in action to provide the breath support you need throughout the exhalation.
It's easier to get breath when you're singing in your most comfortable range. It's much harder to maintain a very high note than one in the middle of your range. Practice breathing on different tones. Take deep, slow breaths -- remembering to keep your throat open and expand your diaphragm -- and sing a note during the entire exhalation. Start with low notes, as these are more comfortable and help warm up your voice.
The biggest problem beginning singers have with breathing is staying relaxed. You're so worried about the different muscles you need to be engaging that you tense up and the air stops flowing right. Relax your entire body before you begin singing.
You can even practice while laying on your back (but don't stop expanding your lungs and diaphragm). Proper breathing will feel strange and unnatural at first but you'll get used to it. If you're serious about singing, it may be worth it to take some yoga or meditation classes.