There are simple rules of thumb.
Try what the manual says. Install and run on a clean target, user license, the works. Does it work? Did you have to add anything not covered in the manual?
Are all the default control values usable? Or is there something that's wrong, or blank, by default and always has to be changed?
Set every value in the user interface to something other than its default. Can you detect a difference caused by the change? Is it correct? Do them one at a time, or in the smallest sets possible, to make the results clear.
Set every value in the user interface to a second, non-default, value. Change everything at once. Can you detect the difference? Is it correct?
One by one, do something to cause every error message to be generated. Do something similar, but correctly, so that no error message is generated.
All of the above depend on changing a condition, between an "A" case and a "B" case, and that change having a detectable result. Then the "C" case produces another change, another result, and so forth. For 10 tests, you need 11 conditions. Using defaults as much as possible is a good first condition.
By now you've got a list of things to test, that you recorded, and results, that you recorded, and maybe some new bugs. Throw something big and complicated at the solution. Give it a file of 173000 words to sort, paste a Jane Austin novel or some telecommunications standard 100 pages long, a 50MB bitmap graphic, 3 hours of streaming video. Open the performance monitor and get CPU-bound, or I/O bound. For an hour. Check memory use: always increasing? Rises and falls?
Take the list of bugs closed in the last week, month, sprint, etc. Check them. All. Are they really fixed?
Keeping track of what to do, how it worked on what version/release/build/configuration, open and closed bugs, what controls have been set or changed, what data, test files or examples have been used, etc. is all part of Quality world. Keep results as tables in a spread sheet, make version controlled backups / saves snapshots.
Someone writing software, or any one creating anything, has an idea of what they're trying to make. The quality process starts with expectations. Requirements, specifications, rules, or another articulation of what's expected. Then there's the solution, the thing offered to perform, assist, enable or automate what's expected. Then there are tests, operations, examples, inspections, measurements, questionnaires, etc., to relate one or more particular solution(s) to (relevant) expectations. Finally, there's an adjustment, compensation, tuning, correction or other positive action that is hoped to affect the solution(s).
When one writes software, one has a goal of it doing something, and to the extent that's expressed, the behavior can be checked. Hello.exe displays "Hello World" on a screen. "2**150" in the Python interpreter displays, "1427247692705959881058285969449495136382746624L". Etc. For small problems and small solutions, its possible to exhaustively test for expected results. But you wouldn't test a word processor just by typing in some words, or even whole documents. There are limits of do-ability and reason. If you did type in all of "Emma" by Jane Austin, would you have to try her other four novels? "Don Quixote" in Spanish?
Hence an emphasis on expectations. Meeting expectations tells you when the solution is complete. My web search for "Learn Quality Assurance" just returned 46 million potential links, so there's no shortage of opinions. Classic books on the subject (my opinion, worth what you paid for it:) include
"Quality is Free" by Philip Crosby
"Zen and the Art of Motorcycle Maintenance" by Robert Pursig
"Managing the Software Process" by Watts Humphrey
"The Mythical Man Month" by Fred Brooks
"Code Complete" by Steve McConnell
Take 5 minutes to read some of the Amazon reviews of those books and you'll be on your way. Get one or more and read them. They're not boring. Browse ASQ, Dr. Dobbs, Stack Overflow. Above all, just like writing software. DO it. Consider the quality of some software under your control. Does it meet expectation? If so, firm hand-shake and twinkle in the eye. Excellent!. If not, can it be corrected? Move to the next candidate.
I like the Do-Test-Evaluate-Correct loop, but its not a Universal Truth. Pick a process and follow it consciously. Have people try the testing, verification and validation steps described in the language manual they use most frequently. Its right there on their desk, or in their phone's browser.
Look at your expectations. Are they captured in a publicly known place? With revision control? Does anyone use them? Is there any point where the solutions being produced are checked against the expectations they are supposed to be meeting?
Look at your past and current bug reports. (You need a bug tracking system. If you don't have one, start there.) What's the most common catastrophic bug that stops shipment or requires an immediate patch? Whats the most commonly reported customer bug? What's the most common bug that doesn't get fixed?
Take a look at ISO 9000 process rules. Reflect on value to your customers/users. Is there's a "customer value statement" that explains how some change affects the customer's perception of the value of the solution? How about in the requirements?
By "the QA process", you could mean "Quality Assurance", versus "QC", "Quality Control"? You might start with the http://www.ASQ.org web site, where the "American Society for Quality" dodges the question by not specifying "Control" (their old name was "ASQC") or "Assurance".
Quality; alone, "assured" or "controlled", is a big idea with multiple, overlapping definitions and usages. Some will tell you it cannot be measured in degrees- its present or not, no "high quality" or "low quality" for them. Another famous claim is that no definition is satisfactory, so its good to talk about it, but avoid being pinned down in a precise definition. How do you feel about it?