You should unaccept bassir's answer because it only addresses the case in which X and Y are independent, and his statement that Z is Normal may not be true if X and Y are not independent.
If X and Y are independent, then EZ and Var(Z) are as stated by bassir.
If X and Y are jointly Normal (i.e., Bivariate Normal), but not necessarily independent, then Z is a Normal random variable, and EZ is the same as for independent case, and
Var(Z) = 2*σ^2 -2*Cov(X,Y) = 2*σ^2 - 2*(correlation coefficient between X and Y) * σ^2, where Cov(X,Y) is the covariance between X and Y.
If X any Y are not independent, Z does not necessarily even have a Normal distribution. EZ and Var(Z) are per the immediately above case which assumes X and Y are jointly Normal, but in fact, Z may not be Normal. There are several such examples in http://www.amazon.com/Introduction-Probability-Theory-Applications-Vol/dp/0471257095 .