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NancyLebovitz comments on Making your explicit reasoning trustworthy - Less Wrong
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Make every link in a chain of argument explicit. Most of the weirder conclusions I have seen in my own and others' beliefs have come about because they had conflated several different lines of reasoning or have jumped over several steps that appeared "obvious" but that included a mistaken assumption, but were never noticed because they weren't spelled out explicitly.
Also, be very careful about not confusing different meanings of a word, sometimes these can be very subtle so you need to be watchful.
For actually reasoning with an argument, keep it schematic. One of the reasons reading philosophy is so hard, is that it is written in prose. For any but the simplest arguments, though, you need to convert it to schematic form before you can actually reason about it effectively. Like trying to do mathematics or play music from a written description (though not quite that extreme), it just doesn't work well.
I'm interested in examples for the sort of mistakes you're describing.
In general, when you see two people arguing past each other, these kinds of problems are often involved at the root. Two examples that I can give are the problem of "natural rights" and the problem of "authority". The natural rights issue needs a pretty long and involved discussion even to understand but it amounts to a long, convoluted sequence of conflations and assumptions.
The problem of authority is easier to describe, since it amounts to a single major error - authority conflates two distinct ideas - knowledge or expertise and justifiable or legitimate force. The two are necessarily linked in parental authority, but they are distinct ideas that tend to cause misunderstandings and resentment when conflated in institutional academic or state interactions.
A good source for understanding the root idea in a political context is Thomas Sowell's A Conflict of Visions where he points out that people tend to use the same word to mean different things - his main examples are "fairness" and "equality". Those distinct meanings rest on the fact that those words conflate those two (and more) meanings into their definitions - neither side is "misusing" the words - the words themselves, and the fact that most people don't notice the conflation, is the problem.
Basically: A human's guide to words