These are a couple of correlatives that I have continued to have problems with in terms of really understanding how to use them.

For instance, when I look the definitions up I see:

Iu - someone, somebody, any, some

Ĉiu - everyone, everybody, all, each

I think what is unclear to me is how they are used by themselves and how they are used when paired up with other words. For instance, some times I see them by themselves and at other times I see them paired up with a word like "homo" or "persono."

Do they mean someone/somebody (iu) and everyone/everybody (ĉiu) when they stand alone, but any/some or all/each when paired up with another word designating a person?

What would be some examples of each usage?

You might want to take a look at the grammar subsite here at Lernu (at the bottom of the page you can change the language). The tabelvortoj section in that explains that those
 

• beginning with ĉi- are all-encompassing words
  
• beginning with i- are undefined words
  
• ending with -u are individual things
  
• beginning with -o are things

ĉio kaj io

There is a rule in Esperanto that you cannot put nouns and pronouns in nominative right after each other (so chicken soup bowl is not kokaĵo supo bovlo). Since ĉio and io are fundamentally pronouns they take the place of the subject if in nominative, object if in accusative or whatever role shown by a preposition if preceeded by one.

Ĉio usually gets translated to "everything".
 

• Ĉio estas malfacila! : Everything is difficult!
  
• Li faris ĉion, kion li povis : He did everything he could.
  
• Ŝi zorgis prio ĉio : She took care of everything.

Io usually gets translated to "something".
 

• Ĉu io okazis al vi? : Did something happen to you?
  
• Ŝi ricevis ion por manĝi kaj trinki : She got sometrhing to eat and drink.
  
• Ĝi estas la sono de io granda : It is the sound of something big.

Ĉiu kaj iu

The ending -u shows individual things or matters of a group or a set. The group can be given or can be inferred from the context or is assumed to be humans.

Ĉiu usually gets translated to "every, everyone, everybody", i.e. every single element of a group.
 

• Ĉiu homo devas… : Every human must… (explicitly stated group)
  
• Ĉiu devas… : Everyone must… (group assumed to be "all humans")
  
• Estas multaj detaloj, sed vi ne zorgu pri ĉiu! : There are lots of detials, but you should not worry about every single one! (group definied in the preceeding phrase)

For differences between ĉiu and ĉiuj see the grammar section.

Iu usually gets translated to "any, anyone, anybody".
 

• Ĉu estas iu persono en…? : Is there any person in…?
  
• Ĉu estas iu en…? : Is there anyone in…?
  
• Sendu la rakonton al iu gazeto! : Send the story to a newspaper!

A word of warning here. If your mother tongue is a language with an indefinite article, note that the indefinite iu and and iuj are emphasising ones ("anyone will do") and should not be used whenever you would use an indefinite article in your mother tongue.

Okay, so now I'm confused again, because I thought that ĉiu specifically had to do with people, or at least living things. I thought that for non-living things--things such as "details" in your final example sentence--you would use ĉia.

What then is the difference between ĉiu and ĉia?

If you take a look at that grammar page here in Lernu, we have a bunch of beginnings and endings that can go together. Despite these the table words or correlatives are not compound words but simple, non-compound words.

A side note. Common for all beginnings is that they all end in -i and all endings begin with a vowel (which may be followed by a consonant, but not necessarily), so in theory you should make a small pause at that boundary. However depending on the speaker this pause can vary between clearly audible to non-existing.

As I wrote, the ending -o denotes a thing which with the all-encompassing ĉi- usually gets translated to "everything". The ending -u denotes an individual thing (one out of a certain group) which with the all-encompassing ĉi- usually gets translated to "every".

In turn the ending -a denotes quality or type which with the all-encompassing ĉi- usually gets translated to "every kind of" or to "every kinds of" in plural. Note that English requires use of the preposition "of" while de is not required in Esperanto.
 

• La logiko diras, ke ĉia verbo povas havi pasivon, se nur la senco ĝin permesas. : The logic says that every kind of verb can be put into the passive if only it makes sence. (Lingva respondo 111)
  
• Kuko estas ĉia speco de franda pano (ekzemple kun sukero, kun konfitaĵo, fruktoj k.t.p…) : Cake is every kind of sweet bread (for instance with sugar, jam, fruits etc…) (Lingva respondo 47 G)
  
• En ĉiaj aliaj okazoj mi uzas… : In every other kinds of cases I use… (Lingva respondo 84 A)

Here the second sentence has de but that is because of speco, not because of ĉia.

Metsis, I have looked at that grammar page and, while useful, I find that its primary flaw is that is lacks usage examples. It does have a few, but not enough to be overly helpful for me.

When it comes to languages the way I learn best is through seeing a principle first explained, and then seeing multiple examples of that principle in action.

So let me start from the top and see if I am understanding correctly.

Ĉiu

The definition I have is "everyone, everybody, all, each."

Please tell me if these examples are correct:

Everyone will be at the party.
Ĉiu estos ĉe la festo.

Every person is required to register.
Ĉiu persono devas registri.

Each man should learn to read and write.
Ĉiu viro lernu legi kaj skribi.

Every car is red.
Ĉiu aŭto estas ruĝa.

Each book is more than 500 pages long.
Ĉiu libro estas pli ol 500 paĝoj.

Yes, that is the way ĉiu works. It refers to every individual in a group or set. This group can be explicitly stated – like in ĉiu persono or ĉiu aŭto – or is defined by the context:
 

• La plej konataj inter la mamontaj arboj en Kalifornio estas "Tri fratinoj". Ĉiu estas pli ol 90 metrojn alta.

Most often the group is "humans", like in that Ĉiu estos ĉe la festo.

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Some remarks

The Esperanto preposition ĉe is (again) one of those with many meanings, but the general idea is "being very close, at a touching distance". It is even used figuratively for objects or things being close to each other. For instance sidi ĉe tablo, domo ĉe la komenco de la vojo. So my reading of ĉe la festo is "being close to the location of the party, but not participating in it". A better expression would be Ĉiu ĉeestos en la festo, see ĉeesti in PIV.

In "Every person is required to register" the persons both need to do something (are subjects) and are the targets of the action (are objects). Such verbs are called reflexive verbs. While in English a verb can be both ("I open the door" vs. "The door opens"), in Esperanto this reflexivity must be explicitly marked, thus Ĉiu persono devas registrigi sin or Ĉiu persono devas registriĝi (for reasons why registrigi, see registri in PIV).

Ĉiu viro lernu legi kaj skribi. So only men should?

"Each book is more than 500 pages long." The idiomatic way to express this is with the verb havi, cf. a book has more than 500 pages: Ĉiu libro havas pli ol 500 paĝojn (accusative of object). Of course if you want to use the verb esti, that is fine, but then you have to add what you are measuring with those 500 pages, i.e. the length. This is done with an adjective longa. To connect the measure and what is measured you need to use a construction called "accusative of measure": Ĉiu libro estas pli ol 500 paĝojn longa. See the grammar section in Lernu.