the Greeks are just the derivatives (I know, confusing because derivative means 2 things):

if V is the option price, S the stock price, and \sigma the volatility, then

Delta =\frac{\partial V}{\partial S}

Gamma =\frac{\partial\;\textrm{Delta}}{\partial S}=\frac{\partial^{2}V}{\partial S^{2}}

Vega =\frac{\partial V}{\partial\sigma}

If you remember put-call parity,

long call and short put = long stock and short bond

call - put =S-Xe^{-r(T-t)}

If you differentiate put-call parity with respect to stock price, you find

Delta(call)-Delta(put)=1.

If you differentiate a second time with respect to stock price, you find

Gamma(call)=Gamma(put)

Similarly, if you differentiate put-call parity with respect to volatility, you find

Vega(call)=Vega(put).

In the Black-Scholes model, you have a formula for the value of a call and of a put, and you can differentiate those formulae to find the Greeks.

Gamma =\frac{1}{\sigma S\sqrt{2\pi(T-t)}}\exp\left[-\frac{1}{2\sigma^{2}(T-t)}\left(\log(S/X)+(r+\frac{1}{2}\sigma^2)(T-t)\right)^2\right]

Gamma is the same for the call and the put.

r is interest rate, \sigma is volatility, t is time, with option expiry at t=T

When stock price is zero, Gamma is zero.

For stock price S>0, Gamma is >0.

Hence `non-negativeâ€™ which means it can be either positive or zero but not negative,

Under the Black-Scholes model, if the stock price is zero, it remains zero. But if the stock price is >0 it will never become zero.

Gamma is the derivative of Delta with respect to stock price

For stock price >0, Gamma will be positive which means Delta will always increase as stock price increases for S>0.

As you say, for the call, Delta goes from 0 (when S=0) to 1 (in the limit S\to\infty)

and for the put, Delta goes from -1 (when S=0) to 0 (in the limit S\to\infty)

If you work out Vega (derivative with respect to volatility),

Vega =\frac{S\sqrt{(T-t)}}{\sqrt{2\pi}}\exp\left[-\frac{1}{2\sigma^{2}(T-t)}\left(\log(S/X)+(r+\frac{1}{2}\sigma^2)(T-t)\right)^2\right]

so that Vega=Gamma \times S^{2}\sigma(T-t)

Yes, the formulae are alike, but Vega and Gamma are different things.

If you graph them, Vega and Gamma both look like skewed Normal Distributions.

Vega will be larger than Gamma as S\to\infty

Gamma will be larger than Vega as S\to 0