I am emailing my supervisor and I want to let her now that I am taking a training that happens every week at the same time (like a series of trainings).
It sounds to me like you are having "regular" training. You could also say "weekly training"
Recurring isn't quite the right word. If the training recurs, it sounds like you are being trained on the same thing. Recurrent training similarly suggests "the same event happening over and over." A recurrent dream is when you have the same dream repeatedly.