How to standardize my data (Univariate Time Series Forecasting using Keras LSTM)?
Welcome kakarotto! The first thing, standarize with respect to the training set only, then use the statistics to standarize the other sets. As a rule of thumb, we should keep in mind that the test set does not exits while training... (although we have access to it) When training, if the network does not see the test set we (the humans) do either… 

If the test set was well selected, and so the whole dataset, the training set and the test set should come from the same probability density function and, if enough data is provided, the statistics you get from the training set would be close to the statistics of the pdf (assuming unbiased estimators for these statistics, that should be the case for the mean and standard deviation)

Based on my experience, only standarize the input data. With the output data, I usually scale it (min-max scaler) if I know the boundaries (it they even exist…). Another usefull trick that usually worked for me its to not feed the net with the input data rather with its relative differences, for example:

$X = (0, X_1 - X_0, X_2 - X_0,\dots,X_m-X_0)$