If you want to know when you can omit that it is essential that you first understand the different functions of the word. In your examples, the that in the following sentences introduces an object clause:
I recommend that you take my advice.
I know that you are correct.
Similar examples are:
I hope that you have a happy Christmas.
I understand that you meant well.
In your other sentences the that introduces a relative clause (and could be replaced by 'which' in each case). Your examples:
I bought the book that (which) is required for this course.
The report that (which) was approved by the board was written by
We are studying advertisement strategies that (which) other companies
use to recruit minorities.
Having clarified this fundamental difference in the function of that, we can now address the question of when that can be omitted. In the case of that introducing an object clause, in most cases it can be omitted - as it can be in your two example sentences. Note, however, that we do not usually omit that after intransitive verbs:
?He emailed he had sent the money.
?She whispered she was not feeling very well.
We also do not usually omit that after nouns:
?I don't believe his claim he was sick.
?Do you agree with the view what he did was wrong.
Turning to the second function of that (introducing a relative clause), whether it can be omitted depends on whether that is the subject or object of the clause. If it is the subject it cannot be omitted (but see below for passive constructions). If it is the object, it can.
Subject: Philatelists are people that collect stamps.
Object: Show me the stamps (that) you bought last week.
Object: We are studying advertisement strategies (that) other
companies use to recruit minorities.
Finally, note that in your other two relative clause sentences, you cannot omit just the that but must also omit the finite verb:
I bought the book (that is) required for this course.
The report (that was) approved by the board was written by Susan.
What we have here is a passive construction which permits such elision.