Hi, Paul,
Thank you for your question.
Your first sentence (I’ll teach you.) is written in the simple future tense. It indicates that the teaching will be done for a short (relative term) period of time. For example, maybe you will teach me how to brush my teeth, which can be done in a matter of minutes.
Your second sentence (I'll be teaching you.) is written in the future progressive a/k/a future continuous tense. We use this form to indicate that the action will be ongoing for a long (relative term) length of time. For example, you might have been assigned as my biology professor for the upcoming fall semester, a duty that will take several months to fulfill.
By the way, a second way you could have written your future progressive/continuous statement is this: I am going to be teaching you. Both forms are acceptable.
I hope you find my explanation helpful. Should you need additional help with grammar-related issues, please don’t hesitate to schedule a private session with me.
