After you have the proper posture, the next step is to learn to breathe correctly. Breathe deeply. As you breathe imagine you are smelling a rose and feel your lower back expand. This will help you gather all the air necessary to hold long notes. This will not happen right away but with time you will be doing it. First start singing the vowels. One good way to understand how to sing correctly is to feel as if you are yawning. If you notice, your throat will expand and allow the air to flow freely. Practice scales from high to low, and low to high without hurting yourself. You want to start easy and build as you practice. If a note is too high for you, don't try it. This will warm up your larynx and allow you to stretch your vocal cords.
Professional singing is done from the diaphragm, so a singer's midriff should move, not their shoulders as though they were panting. One simple exercise is to lie on the ground and place a light book or box on the midriff. This way, you can see the box moving up and down when you are singing correctly.
Once you have been singing for a while, you may want to "improve your range". This means that you wish to be able to hit more notes correctly and without hurting yourself. To do this, simply go through scales as normal, but when you get to the highest note you can hit, flip your head upside down and try to hit that elusive high note. Chances are you will be able to do it!
If you want to hit a low note, do the same as described above except flip your head back instead of forward.