Singing in a different key.?

When I asked my friend for an honest opinion about my singing voice,
she told me that I'm off tune in some parts of the songs I sing. What
she meant was that, I usually sing in a different key than the
original. Or like I sing it in a lower pitch.

I think this isn't really a bad thing considering the vocal range
difference of people but I'm no professional so I don't know for sure.
What I mean to say is that, say the song starts off with a C note, I
start it off with an A or B or whatever, but the transitioning from
note to note is pretty much the same. Do I really need to start on a C
note if I want to sound good?

Also, how am I supposed to know if I'm singing from my diaphragm? I
mean, without having to go to vocal class. I just want to sound fairly
good since I play guitar and what's a guitarist who can't sing...