Hi @menardmam:
Let’s try with what I wrote in the question :
$cat->cat_name. How can I know thatcat_nameis a method that return something from$catfromget_the_category()function?
Frankly the best way to know an answer like that is to use a debugging IDE and to trace through the WordPress core source code. Seriously; the source code is the only documentation that is sure to have been updated, and the results shown in the debugger are the only documentation that is sure to be exactly correct. I use PhpStorm+Zend Debugger but others use NetBeans, others Eclipse, and there are several other PHP IDEs with debuggers too.
Next, you can “google” for your specific questions by including site:codex.wordpress.org as part of your search term and/or search for it here on this site.
As a third and often best way, ask a specific question here on this site. This site works best when you ask specific “How-To” questions with a title that matches your question and with details about your question, and especially your use-case when applicable. Think about it this way:
If someone googles to find the answer you are after, would google ever match their search terms with the title of your question? And when they do find your question would it be very clear to them what question you were asking?
If you can answer “Yes” to both of those then chances are very good you’ll get a great answer to your specific question here and you will be likely to get it quickly.