One way is to look for the subject. If that comes before the subject, then you can omit that.
When that isn't required
"He told me (that) I should leave"
"They did not like the work (that) we had done"
When that is required
"I am looking for a book that is easy to read."
Some people would say (that) using "that" is formal because the closer the language is to its orginal form, the more formal it is believed to be.
Yes, it's a good rule of thumb to keep that in a sentence, especially if you don't yet know its patterns.
There's more here:
You can omit that
in a relative clause when the subject of the clause is different from the word or phrase the clause refers to. Thus, you can say either the book that I was reading
or the book I was reading.
You can also omit that
when it introduces a subordinate clause: I think we should try again.
You should not omit that,
however, when the subordinate clause begins with an adverbial phrase or anything other than the subject: She said that under no circumstances would she allow us to skip the meeting. The book argues that eventually the housing supply will increase.
This last sentence would be ambiguous if that
were omitted, since the adverb eventually
could then be construed as modifying either argues
or will increase.
Source: § 62. that. 1. Grammar. The American Heritage Book of English Usage. 1996