Given two different Gaussians (two different means and standard deviations,) how do I add them together to make a single gaussian?
The sum of two Gaussian distributions is Gaussian with mean $\displaystyle \mu_1 + \mu_2$ and variance $\displaystyle \sigma_1^2 + \sigma_2^2$. Assuming independence, that is.
See
Normal distribution - Wikipedia, the free encyclopedia
under "Properties".