First, the Cullen & Frey plot shows $\text{skewness}^2$ versus kurtosis, which are third and forth standardized moments, so standardizing the data (in the usual sense of subtracting mean and dividing by standard deviation) makes no difference.
You didn't tell us much about your modeling goals, neither about your data (sample size? What does it represent? ...) so we cannot say much about "what you should do downstream", but at least, you should do as @glen_b says and understand your tools better (and tell us more about your problem.
But your plot: The big blue dot represents your data, and it stays away from the points representing normal and exponential distributions ..., and also away from the lines representing gamma and lognormal. Projecting vertically to those lines, you can see that you have higher kurtosis than a gamma/lognormal of the same skewness. So maybe try some extended gamma, or log-skewnormal?
But, your plot misses some important information for judging this: The uncertainty in the sample estimates of skewness/kurtosis. That you can evaluate by bootstrapping, see my answer here for example and code.
Normalized data
