Confused on normality assumption I know that the sampling distribution of the mean can be assumed to be normal if N>30, but does this have an implication on the "30" itself (the sample data)? 
I have three different time series with N=500 (well beyond 30) each and I want to test for equality of their means. Does the normality of the mean of the sampling distribution mean that I can assume the normality of the time series data I have and use a parametric test? For example Welch ANOVA?
 A: It is not correct to say that a sample size of 30 or so makes the central limit theorem apply.  Take for example a sample from a log-normal distribution with n=50,000.  The CLT when used to construct a confidence interval for the unknown mean yields very inaccurate limits.
Use a method that does not assume normality, e.g. a nonparametric test.
But note that none of this applies directly to time series when the multiple observations within a series are correlated.
A: No, by the central limit theorem the sampling distribution of the mean approaches normality regardless of the form of the parent (with a couple rare exceptions). While you compute only 1 sample mean per group, that point estimate is an exemplar of a family of possible sample means you might compute with infinite resources. That distribution is normal. 
A: Normality is something we commonly assume when we conduct a hypothesis test or fit a model. A common rule of thumb is that if your sample size is greater than 30 the central limit theorem probably applies and you could use a test that assumes normality. This is a rule of thumb only, and is often violated. If there is ever a magic cut-off number in statistics be wary of it (another common one is for comparing ratio of variances)
You can check the normality assumption of your data with all sorts of tests or graphical procedures (such as qqplots). You should check the assumptions of the test you want to do (anova in this case) and if they are not violated then you can proceed and trust that your p-value etc are actually meaningful.
This question here has a list of the assumptions and an interesting discussion of what normality we are interested in (normality of the residuals or normality of the individual groups).
ANOVA assumption normality/normal distribution of residuals
