Like the above poster have mentioned, having a wide vocal range does not mean you can sing well. Singing is not just about hitting the notes on key.
You should never strain your throat to go higher or lower otherwise you risk permanent damage to your voice. Before you start singing anything, identify your singing voice with the help of an instrument like a keyboard/piano/guitar/etc. C4 is common notation for the middle C. Find the limits of your voice in the modal register (simply put: your 'normal' voice), no falsetto, no whistle, no vocal fry.
If you are a male, you could be a:
Bass: E2-E4 (usually a true bass can even go lower than C2)
Baritone: G2-G4
Tenor: C3-C5
If you are a female, you could be a:
Contralto: F3-F5
Mezzo-Soprano A3-A5
Soprano: C4-C6
Note that above are just expected vocal ranges of the different voices. Many people often can go higher or lower than these stated ranges but still classified under the same type of voice. For example a baritone can go as low as C2 but as high as A4, but he is still classified a baritone (actually bass-baritone) because of his timbre, vocal weight and comfortable range.
To widen your vocal range, first learn how to sing in your comfortable range to grasp the techniques, then attempt the extreme ends. But, for instance, if you are a baritone, you can never sing like a true tenor, and vice-versa. So for starters, you should never attempt to sing like another voice type, or you will risk vocal damage as a result of straining.
Finally, it is a common misconception to think that just because someone can sing higher, he/she has a bigger range of voice. This is untrue because most of the time, the number of high notes Person A can do that Person B can't, is roughly equal to the number of low notes Person B can do that Person A can't. So in that case, both A and B have the same vocal range, but not the same vocal type.