Learn English – Use of three tenses in one sentence

I think that you'll be fine with any of the three in terms of clarity of meaning, though I'll discourage the past perfect in this case.

To my eye the issue seems more grounded in the first clause of the sentence: I usually have to read over, which establishes that the speaker engages in this behavior exclusively in the present (if not on a regular, ongoing basis). I think that present perfect and simple past are both appropriate here.

The relevance of the material that was taught is still in effect at the time of any given review, so the past perfect seems undesirable to me. It is also the case that the past perfect suggests that the pattern is not ongoing, which is at odds with the first clause's suggestion of an activity that still occurs. Such a reading would suggest that the speaker regularly reviews things taught in a specific period of the past, such as reviewing his/her own course notes from a specific class taken years ago, repeatedly. Conjugation in the second clause aside, I would not find this to be a natural reading of the sentence as constructed.

As for the other two options, with no context beyond what was presented in the OP I think that they are functionally interchangeable. Presumably each "unit" of teaching is what is being reviewed, whether that's by topic, course, day, semester, individual lecture or whatever. Each review is paired with some amount of teaching, but that the groupings are appropriate is implied by the larger sentence.

So, whatever unit is being considered, that material definitively was taught in the past-- presumably the speaker would not be reviewing planned material for an upcoming lesson at the same time as reviewing material actually delivered (if this is the intended meaning, the sentence itself does not suggest so). At the same time, it remains the case in the present that that material has been taught, and the review of that past action implies that that action is still relevant at the time of review.

TL;DR: My read of the sentence favors the simple past, as each "unit" of teaching is discrete and occurred only in the past with regard to any given review, and its relevance to the present exists only within the context of the review itself. The present perfect is also technically correct and appropriate but adds no useful information beyond what the simple past supplies. Past perfect may not be appropriate because it suggests a different meaning than the other two options suggest, leading me to believe that this is not the intended meaning.

The short answer is that in good writing, you shouldn't switch tenses the way you did in your examples. Inserting the narrative present into a narrative that is otherwise in past tense has to be handled very carefully with good transitions between. Usually, you want something to indicate that you are now going to tell a story within a story of sorts. And after you switch to the present tense, you want to stick with it for the entire story. For example:

I had the strangest experience with my mother the other day. We're walking down the street and she's holding my hand tighter than usual. I don't know why. An expensive car pulls up right next to us, and I look at it. My mother looks at me looking at it and slaps me.

The first sentence sets the stage for telling the story about your experience. Then the experience itself is related in present tense to give it a sense of immediacy. Some writers would insist that you shouldn't make a tense switch like this one, either, but in my opinion it can sometimes be an effective tool (when used sparingly). You might, especially, see it used to relate dreams within a story, for example.

In spoken language, however, people mix up tenses all the time, and I can easily imagine someone mixing their tenses in speaking the way you did in your mother story (although not the way you did in your bus one—that one's too extreme even for speech). But it's not necessarily purposeful, it's just a question of mouth moving faster than brain—and possibly of the person telling the story switching in and out of a personal sense of immediacy. For example, the person is possibly living in the memory of the story when saying "she's been holding my hand tighter than usual" and then coming back to the true present to reflect on it with "I didn't know why she did that."

The reason the bus example feels too extreme is simply that it's just too short to support the tense change. There's isn't a separate subnarrative there. It's only two sentences, and each is in a different tense.