What simplest set of rules to remember in order to get commas right?

Use them to separate the parts of sentences, which are referring to different or tangential things.

For example, when clarifying a point or mentioning an extra piece of information, such as a condition or an example like this one.

Whenever you would pause in natural flow of speech.

That should get you most of the non-mandatory-but-extremely-beneficial-for-the-reader commas, and the other instances have rules, such as “always before ‘which’, never before ‘that’”, etc.

Unfortunately, there really isn’t—since advanced comma usage is remarkably related to whatever style guidelines a specific publication uses. Eats, Shoots, and Leaves has a great section about the subjectivity of comma usage.

As with all punctuation, you want make your written text as clear and precise as you can for the reader.

This is for a semicolon but it goes into using commas and periods also.

Technically, you need a comma at the end of every clause, whenever that’s not also the end of the whole sentence, and you also need commas after items in a list (“apples, pears, peaches and lichees”).
But it’s also a matter of style. Using a comma makes two clauses seem less connected, and you can sometimes use that effect by leaving a comma out to convey a subtly different message.
As a general rule of thumb, the longer the sentence, the more commas it should have.
Something to do with limited working memory and the reader’s brain needing a break every now and then.

I don’t blame you if figuring out when something is a clause is only even more confusing, though.
I think the best approach is just to get an intuitive feel for when to use commas, by reading and paying attention to where other people put them. Language is used subconsciously, after all. Explicit usage rules are just your training wheels.

I don’t know of any single simple rule. Rather, there are rules for various kinds of cases, purposes, and situations.

This is not so difficult, really; we learn to discriminate among situations and apply the correct rules for all kinds of things. Mastering the comma is much less complicated than, for instance, learning how to cook (which things do you boil, fry, grill, roast, etc., and for how long, and with what preparation and what serving method?), learning the rules and strategies and techniques of some video game, reading music and playing an instrument, writing software code (which does not leave room for approximations and gray areas), and many other things that we tackle in the course of ordinary life. They don’t come naturally; they all take some effort, attention, and practice; and they do, for the most part, make sense.

I like @Jeruba‘s cooking answer – to extend the analogy, the comma is used in some cases to determine the procession of thoughts or ideas.

unwrap the chicken rub salt on the skin scald the skin rub gralic butter on it put it in the oven at 220 degrees for… etc.

Where would you put the commas in the above gibberish? I would put them where the ideas separate, personally.

However, that is a drastic simplification, and I’m sure there are exceptions.

Use a comma when you would pause if you were speaking instead of writing.

Other than that, as @andrew said, it’s a matter of personal style.

@bob_ Well done. That is as close to a rule as we’re likely to get.

In my opinion that rule ^^^ is more misleading than otherwise.

To reiterate my point, which seems to have been misunderstood, the rules are there and can be learned with a reasonable investment of effort and practice, just as people learn the rules of cooking, programming, gaming, etc. The fact that we all use the language does not mean that all ways of using it are equally legitimate from the point of view of established standards. I’m not here to say that everyone must comply with schoolbook English; I’m replying to the OP that a person who can learn other rules and systems can learn how to use commas if he puts his mind to it, but he shouldn’t expect it to just come automatically.

@andrew didn’t say it’s a matter of personal style. He referred to a publisher’s style guide, which might be an in-house manual or a standard reference such as Chicago Manual of Style. The function of a style guide is to establish consistent practice for a publisher by specifying which form to follow among acceptable alternatives and prescribe how to handle matters that are not grammatical, such as list elements, use of italics, and so on.

I was taught I should place a comma where I would naturally take a breath… either in or out.

I agree with @Jeruba. This “put a comma where you’d pause speaking” rule is pretty useless and misleading in practise.
Look up some sentences with commas in them, then read them aloud, and listen to yourself, and see whether you pause at every comma (without doing it on purpose).

I for one notice I often talk right over where the comma would be without pausing at all. Because it would often actually be a bit awkward in real life and when you listen to someone talking you can handle long uninterrupted strings of text much better than when you read them anyway because intonation and timing and other non-verbal aspects of the sentence’s presentation can give you hints about the syntax of the sentence whereas when you’re reading it it takes more effort to reconstruct the intended syntax in your mind. Because you only have the words themselves.

I also notice, I, um, often… pause when, er, when there shouldn’t really be a comma but I just, take a moment to… figure out what else I want to say, in this sentence.

Written text is not just spoken text on paper. Written text and spoken text are very different media, and they both have their own rules.

Well, I didn’t say it was a perfect rule.

@Jeruba I know what a style guide is, thank you (I recommend this one). I guess I should have said that one should follow the style dictated by the publication one is writing for, as @andrew said, or simply follow one’s own style when the text is not intended for a publication, or if said publication lacks a manual.

@Fyrius If you were to make an exact transcription of what you’re saying, including “ums” and “erms”, you should indeed use commas. I, um, think it would look, like, kind of weird, but hey, it was your idea.

Essentially punctuation supports meaning and should be used meaningfully. That’s why there are rules and accepted practices with varying degrees of formality. The “pause” rule is something that elementary-school teachers are apt to say as a first cut at explaining the use of commas; this should be supplanted by a more advanced understanding later on in the educational process.

When you’re an accomplished chef, whatever you decide to do in preparing a dish is fine because it is backed by experience and a deep understanding that confers authority on your choices even if you depart from standard practice. If you’re a novice in the kitchen, though, better follow the recipe.

Two people can break the same rule, and one does it out of personal style while the other does it out of ignorance. I don’t think most people are up to making their own rules for commas.

@Jeruba I agree, which is why I answered with the “pause” rule: the OP asked for the simplest set of rules. What’s simpler than elementary school?

Yeah, really the best way to learn is just to read thoroughly for years. Older writing is especially educative, as it often exhibits more complicated sentence structures, and so provides a more varied and comprehensive demonstration of grammar usage.

Keep reading Victorian fiction, eventually you’ll even be thinking in 100-word, sub-sub-clause’d sentences…

Or study law, hahaha, then you’ll know commas…

@bob_, it does meet the “simple” criterion, all right. But here’s an even simpler one: put a comma in the middle of every sentence. Simpler still: don’t use any commas at all. As far as simplicity is concerned, you can’t beat that.

But then there’s the other half of the question: “in order to get commas right.” That’s the problem with all three of those rules. They won’t help you get commas right.

@Jeruba I disagree. The comma-when-pausing rule helps you get some commas right, which is more than I can say about your “I don’t know of any single simple rule” quip. As I said before, it’s not a perfect rule, because there’s a trade-off between simplicity and effectiveness.

Well, that’s true, @bob_, you will get some commas right that way. But if that’s the only tool in your toolkit, you won’t know which ones they are or why they’re right.

@Jeruba True, but, again, you’re missing my point. Getting some commas right is better than not getting any commas right because the rules are “too complicated”. Don’t worry about my toolkit, though, it’s @timtrueman the one who’s asking for tools.

I understand, @bob_. That was a generic “you” and not you you. I have a healthy respect for your toolkit.

As for my point, it was that learning how to use commas properly isn’t that hard and is worth a little effort. I never said it was too complicated. I said it wasn’t. You can gain a lot of competence with less of an investment than it takes to wrote a simple program.