show that if the two samples come from normal populations, then sample mean1 - sample mean2 is a random variable having a normal distribution with the mean1 - mean2 and variance1/n1 + variance2/n2.
You should start by finding the distribution of the sample mean $\displaystyle \overline{X}_1$ (and hence $\displaystyle \overline{X}_2$).
I think this is done somewhere in MHF - you will have to search for it. Alternatively, I'm sure it will be somewhere on the web and certainly many textbooks do it. And you could always have a go at doing it yourself. If you're truly stuck with this, please say so (after you've made an effort to try and find it).
Then use the usual formulae for finding $\displaystyle E\left( \overline{X}_1 - \overline{X}_2\right) $ and $\displaystyle Var\left( \overline{X}_1 - \overline{X}_2\right) $.