In Keras, we can return the output of model.fit to a history as follows:
history = model.fit(X_train, y_train,
batch_size
A history objects has a history field is a dictionary which helds different training metrics spanned across every training epoch. So e.g. history.history['loss'][99] will return a loss of your model in a 100th epoch of training. In order to save that you could pickle this dictionary or simple save different lists from this dictionary to appropriate file.