Learn Hardcore Esperanto: Mi legas la gazeton, kaj poste mi sendas retpoŝton al mia amiko. - I read the newspaper, and afterwards I send an email to my friend.

The -n ending marks the direct object of the verb in Esperanto.
Here mi (I) is the subject, legas (read) is the verb, and la gazeton (the newspaper) is what is being read, so it gets -n.
Any normal noun that is directly receiving the action of the verb takes -n:

Mi trinkas akvon. – I drink water.
Mi legas la gazeton. – I read the newspaper.

Why is there la before gazeton, but no article before retpoŝton?

La is the definite article, like the in English.

la gazeton = the newspaper (a specific one, or newspapers in general as a known thing)
retpoŝton (without la) = an email / some email (not specified which one)

In Esperanto there is no word for “a/an”; the lack of la usually means something like a or some.
So Mi legas la gazeton vs Mi sendas retpoŝton mirrors I read the newspaper vs I send an email.

Why does retpoŝto also get -n (retpoŝton)?

Again, -n marks the direct object.
In the second clause, mi is still the subject, sendas is the verb, and retpoŝton is what is being sent, so it gets -n.
Both gazeton and retpoŝton are direct objects in their own clauses, so both need -n.

Why is it al mia amiko, and not al mian amikon?

In Esperanto, most prepositions (including al) do not use the accusative -n.
The preposition al already shows the role “to / toward”, so you keep the noun in its basic form:

al mia amiko (to my friend), not al mian amikon.

You only add -n after a preposition when that preposition is being used to show direction in space or time, like sur la tablon (onto the table) instead of just sur la tablo (on the table). With al, you normally never add -n.

What is the difference between mia amiko and la amiko?

mia amiko = my friend (possessive, someone who belongs to “me”)
la amiko = the friend (a specific friend already known from context)

You can even combine them: la mia amiko is unusual; normally you’d just say mia amiko.
In this sentence, mia amiko is natural because we are talking about my particular friend.

What does poste add? Could I just say Mi legas la gazeton, kaj mi sendas retpoŝton…?

Poste means later / afterwards / then in a sequence of actions.

Mi legas la gazeton, kaj poste mi sendas retpoŝton… = I read the newspaper, and then I send an email…

If you drop poste, it becomes simply:

Mi legas la gazeton, kaj mi sendas retpoŝton al mia amiko.
This still makes sense, but it does not explicitly emphasize the order; poste makes the “first this, then that” relationship clear.

What is the difference between post and poste?

post is a preposition: “after” (followed by a noun or phrase).
- post la tagmanĝo – after lunch
poste is an adverb: “afterwards, then, later”.
- Mi manĝas, poste mi laboras. – I eat, then I work.

In the sentence you gave, you need the adverb form poste, because it modifies the whole clause mi sendas retpoŝton.

Why are the verb forms legas and sendas the same for mi? Shouldn’t they change like in English?

In Esperanto, verbs do not change with the subject (no I read / he reads type changes).
The present tense ending is always -as, for all persons:

mi legas, vi legas, li/ŝi legas, ni legas, ili legas.

So legas = read / am reading / is reading / are reading, depending on the subject and context.
Same with sendas.

Does legas mean “read” or “am reading”? What about sendas?

Esperanto’s simple present -as covers both English simple present and present continuous.
So mi legas can mean:

I read (habitually), or
I am reading (right now),

depending on context.
Likewise, mi sendas retpoŝton can be I send an email (as part of a routine) or I am sending an email (right now).

Do I really need to repeat mi in …kaj poste mi sendas…? Could I say …kaj poste sendas…?

You can omit the second mi if the subject is clearly the same:

Mi legas la gazeton, kaj poste sendas retpoŝton al mia amiko.

This is grammatically fine and sounds normal.
Repeating mi is also correct and sometimes slightly clearer, especially for beginners or in longer sentences. Both versions are acceptable.

Can I move poste or al mia amiko to other places in the sentence?

Yes, Esperanto word order is fairly flexible as long as it stays clear. For example, you could say:

Mi legas la gazeton, kaj mi poste sendas retpoŝton al mia amiko.
Mi legas la gazeton, kaj poste al mia amiko mi sendas retpoŝton. (more emphatic, a bit stylized)

The neutral, most typical version is close to what you already have:
Mi legas la gazeton, kaj poste mi sendas retpoŝton al mia amiko.

When would I use plurals like gazetojn or retpoŝtojn?

Add -j for plural, and keep -n if it is still a direct object:

Mi legas la gazetojn. – I read the newspapers.
Mi sendas retpoŝtojn al miaj amikoj. – I send emails to my friends.

So the pattern is:

singular object: gazeton, retpoŝton
plural object: gazetojn, retpoŝtojn.

What exactly is retpoŝto made of? Why does it mean “email”?

Retpoŝto is a compound word:

ret- from reto = net, network (here: the internet)
poŝto = mail, post

So retpoŝto literally means net-mail / email.
In practice, retpoŝto can mean email in general, and retpoŝton (with -n) here is understood as an email message. If you want to be very explicit about single messages, many speakers also use retmesaĝo (net-message).