You seem to be confusing between subject pronouns (io, tu, lui, noi...) and direct/indirect pronouns (mi, ti, li/gli, ci...), the ones that are explained in the lesson.
In English, object pronouns behave differently than Italian. Let's take I love you as an example. This you is the same as the subject pronoun you, but it actually changes in Italian: io (subject pronoun) ti (object) amo.
To/for you is still translated with ti in Italian:
I give you a gift = I give a gift to you
Ti faccio un regalo.
Ti can only be used with a single person (singular you), it's not plural.
I follow you = ti seguo
I don't understand you = non ti capisco
I will let you know = ti farò sapere
I ask (to) you a favour = ti chiedo un favore
Vi, however, is used to address a group of people in both informal and formal context (Le (as Loro) is still contemplated in classic grammar books, but it's considered outdated nowadays where people use Vi instead). Let's reuse the same example as before:
I give a gift to you all = Vi faccio un regalo
How do you all (yourselves) feel today? = come vi sentite oggi?
"Do you feel yourselves ready?" = vi sentite pronti?
Non vi dico cosa è successo! = "I'm not telling you (all) what has happened!"
Ci is different from noi because you can't use it as a subject. The same happens in English as well, because you can't use us as the subject of a sentence.
Noi siamo felici = We are happy
Ci siamo felici, us are happy
Consider this example:
Lui ci dà un regalo = he gives us a present
Lui (subject pronoun) ci (to us) dà (gives) un regalo.
Ci vediamo domani!
"We see each other tomorrow!"
We don't understand each other = non ci capiamo
Can you do (to) us a favor? = puoi farci un favore?
Now for the polite sentences.
Vi is used in the plural, La, Le in the singular (the same that are used for her and to/for her).
La is the direct object pronoun. I see You = La vedo. (Alternative translation = I see her)
Le is indirect. I give a present to You = Le do un regalo. The object here is regalo, so You is indirect. (Alternative translation = I give her a present)
To them is either translated as loro (no case letter) or gli, where loro is mainly a literary form. Gli matches the pronoun for him/to him.
Gli dico qualcosa, without context, could either be I say something to him or I say something to them.
For the difference between te/ti, you might find this useful: https://members.rocketlanguages.com/members/forum/italian-grammar/ti-vs-te
This topic is huge! If you have any other questions, don't hesitate to ask :)