Are there any simple rules for choosing the definite vs. indefinite (vs. none) article?

I can’t for the life of me figure out where to use a and where to use the — and where there is no article at all. Is there a simple rule of thumb to memorize?

The standard rule you always hear:

“If a person knows which item you are talking about then use "the"

. . . doesn’t clear things up for me, as I have no idea whether or not they know.

Well, if you insist on the rule being simple, here you are:

a = some, any
the = this, that

Two simple examples. Note that you just wrote "...if a person knows which item you are talking about...". You didn't write "...if the person knows...". And that's correct, because you are not pointing to this or that person, you are talking about any person in general.

On the other hand, my answer starts with "if you insist on the rule being simple". That's because you asked for a rule (= any rule), and I am now talking about that rule. We are talking about the same thing.

Now, I can't think of a (= any) simple rule of thumb when not to use an (= any) article at all, but here are some suggestions:

Don't put an (= any) article before a (= any) noun if the (= that) noun is preceded by:
- a number
- a possessive adjective ("my", "his", "our"...)
- a "no", "some" or "any"
- a "this", "that", "these" or "those"

Examples:

Give me a chair! (= any chair you like)

Give me the chair! (= this chair)

Give me that chair! (no article, you already specified which chair you mean)

Give me my chair! (no article)

Give me five chairs! (no article)

Give me some chairs! (no article)

Give me the chairs! (= these chairs)

Give me these chairs! (no article)

Give me a reason to hit you! (= any reason will do)

Give me no reason to hit you! (no article because of "no")

Give me no reason to hit you with a chair! (= any chair)

Give me no reason to hit you with the chair! (= this chair)

There isn't a simple rule, as the definite and the indefinite article are used in different cases.

Eat an apple before going to the cinema.

The sentence is not referring to a particular apple; it can be a random one.
Follow the President.

The sentence is not referring to a random president.

The difference is not between talking of something the other person knows, or not.
In the second example, the person I am talking to could know all the presidents present in the meeting, but he will understand to which president I am referring.

Albert taught himself to play the violin.
Worry about the future.

In this case, the article is used to make a generalized reference to something rather than identifying a particular instance.
They placed the African elephant on their endangered list.

African elephant is referring to the whole species, not to a single elephant.
She is a McFry.

The sentence is referring to a member of the family McFry.

I can’t for the life of me figure out where to use a and where to use the — and where there is no article at all. Is there a simple rule of thumb to memorize?

The standard rule you always hear:

“If a person knows which item you are talking about then use "the"

. . . doesn’t clear things up for me, as I have no idea whether or not they know.

That's fine. All you need to do is tweak this guideline (it is not a rule) to

“If you think/assume a person knows which item you are talking about then use "the" (or some other definite determiner)

See the second sentence in the Wikipedia article on English articles, which is cast similarly to mine (Use of the definite article implies that the speaker assumes the listener knows the identity of the noun's referent (because it is obvious, because it is common knowledge, or because it was mentioned in the same sentence or an earlier sentence)).

In all the following, you assume your listener knows which item (referent) you are talking about, by

(a) context
(b) shared knowledge between you and your listener
(c) knowledge of the world in general.

Examples:
You have just seen a movie with a friend. You say to him:
--Wow, the movie was great wasn't it. a and b

You make lunch arrangements with a friend. You say to him:
--I'll meet you at the cafeteria at 1pm. b (you eat at the same cafeteria each day, or a there is only one cafeteria at which it is possible to eat)

You tell your friend where you left his mail. You say:
--I left your mail under the doormat to the door of your house. This could also be b but c can come into play: most houses have doors, so using the door relies on this knowledge; you could further specify the front door if you felt the need. As for doormat, well that is b, if you both know that the door has a doormat. Or it could be a/c if your friend's wife just bought a doormat and placed it there, your friend can easily figure out which doormat you are referring to.

Your friend wants to know where your wife is. You say:
--She is out walking the dog. Mostly b

There are some idiomatic uses that nevertheless can be explained by this guideline: You have a landline and actully three phones connected to it, so that all three phones ring at once when someone calls. You ask your friend to see who is calling. You say:
--Could you answer the phone? Even though three phones are ringing, no matter which phone you pick up, it will be the phone. It is similar with other things around the house:
--Could you answer the door?

However, there are plenty of idiomatic uses that this guideline does not cover. If your friend asks what your wife is doing and she is reading something, if it is any newspaper, you just say:
--She is reading the newspaper. This is true whether or not you assume your friend can identify which newspaper it is, because the newspaper is idiomatic. It is different from book or magazine, for example, in which you would not use the unless you assume your friend knows which one it is.

On the other hand, a person can use the indefinite article even though he knows you can identify which referent he is talking about:

I. Joe buys a hotdog on the way home. He gets home and he is holding it where you can see it. Joe can say
--Hi. I bought a hotdog on the way home. (Here it seems that the fact that it is 'any old' hotdog trumps the fact that you can easily identify which one it is.)

II. A more subtle case: Joe and his wife Mary are riding home. Joe asks Mary what she wants to do when they get home. Mary can say:

I want to read a book when I get home.

Now, under many analyses, this could be taken to mean 'any old' book, or a book that Joe cannot identify. But in actuality, Mary can say a book even though she knows that Joe knows which one she is talking about. (Say it is the one she reads two pages a night from, each and every night.) For whatever reason, Mary makes an indefinite reference to it, even though she knows Joe knows which one. Why does she do this? Because she can. (English grammar allows this.) And it shows that what I gave at first is only a guideline, not a hard and fast rule.

The opposite is at play too. A person can use the definite article even when he knows his listener cannot identfy which referent he is talking about. Joe can say to Mary:
--Man, the experience I had at work today is not one I would ever like to happen again.
As an opening line in his spoken text, Joe knows that Mary cannot identify which or what experience he is talking about. At this moment this seems to break the guideline I mentioned. But it demonstrates another function of the and that is to bring up a new subject of conversation. Despite many references that will say that a speaker will use a/an to bring up a new topic, we actually use the for this also. And Joe will go on to identify what experience he is talking about. So actually this falls under a context, because the context will make the referent clear as Joe continues his report of what happened at work.