First of all, the mgf of a gamma function is M(t)= (1-βt)^-α ; for β<(1/α)
You forgot to include "t" in the mgf.
To find the expected value, take the derivative of the mgf of the gamma function with respect to "t"(d M(t)/dt), and calculate its value at t=0. This will be E[X] or your expected value as well as the first moment
Then take the second derivative of the mgf of the gamma function with respect to t (d(^2)t/dt^2), and calculate this value at t=0. This will be E[(X)^2] your second moment.
Your variance can be calculated as V(X) = E[(X)^2] - (E[X])^2
or the variance = (the second moment) - (the first moment)^2
You should get : expected value = αβ
Variance = α(β^2)