Must a comma always be inserted between two complete thoughts within a sentence?

No, the comma may be omitted if both complete thoughts are short:

We looked around and we saw no damage. 

However, as the complete thoughts become longer and more complex, a comma is placed before the coordinating conjunction (and, but, or, for, nor, so, yet) to separate the thoughts and facilitate understanding:

The effectiveness of a decision depends upon logical

soundness, and each team member must assume

responsibility for contributing to the discussion

of the possible alternatives. 

We could arrange for lunch to be brought in at noon,

but it might be very difficult to predict exactly

when the speaker will finish her lecture. 

A comma is not used with connecting words like therefore, however, consequently, and so on. Instead, a semicolon is used.