Kavod Rabbi, I am at the beginning of my spiritual journey. A Rabbi suggested that one should try to teach to others what they learn even if they are beginners, he mentioned that is a way for the info to sink into mind. I wonder if I need permission from the Rabbis that I learn to share their lectures with others. Since all the teachings go back to Moshe Rabenu whom by divine order thought Torah all children of Israel for free. I would appreciate your clarification about this matter. Thank you.

No, there is no need for permission, for the Torah is free for all, and belongs to us all. Just that when you cite something you should attribute it to the source or rabbi who taught you that particular thought or fact (Pirkei Avot 6, 6). If it's a lengthy idea, so attribute it to him at the beginning, pointing out that what I'm about to say, I learned from...