Is it recommended to do strength training and speed training at the same time?

Yes. I recommend a multi-modal approach at training, mixing strength & speed workouts. However, that much quality effort must be balanced by appropriate rest periods and supported by good nutrition and sleep. Otherwise, you run the risk of overtraining. I also believe cross-training and preventative measures, such as yoga/pilates/stretching/icing are vital to avoiding injury. Good luck.