How do verbs in Persian work? Right now, I understand that most verbs can be conjugated with regular endings, but I don't understand the relationship between infinitives and their conjugated forms. There also seem to be phrasal verbs like in English.

Here are some examples:
بودن
من هستم
تو هستی
او هست
ما هستیم
شما هستید
اونا هستند

Here, I don't understand how بودن becomes هست

For phrasal verbs, here is an example:
حرف بزنم
as in:
فارسی خیلی خوب نمیتونم حرف بزنم

In this example, it seems like حرف بزنم is phrasal like "meet up" or "hang out" in English.

So how do all of these verbs work? I'm trying to understand so that I can learn more verbs independently.

!خیلی متشکرم

I'm going to put it VERY simple here, there are exception and bigger explanations.

There's a "Bon" (root) for every verb. Bon {root} is the main part of every verb. Some Bons are present tense and others are past tense. Every verb is made by using these Bons and adding a little something to them.
Present tense Bon is the imperative form of the verb, without the extra letters. For example:
گو = say
nobody says "گو", but they say "بگو". this «ب» at the beginning of imperative verbs is essential. But "گو" is the Bon nonetheless.
Past tense Bon is the third-person past form of the verb. For example:
گفت = he or she said
Another way to make past tense Bon is to delete the "ن" at the end of infinitives.

Now when you make verbs, you use the right Bon. Example:
Past form:
Bon: گفت
verbs: گفتم، گفتی، گفت، ...
Present form:
Bon: گو
verbs: می‌گویم، می‌گویی، می‌گوید، ...

Now for your question.
Past Bon: بود
Past verbs: بودم، بودی، بود، ...
But the present form is actually an exception. the present Bon of "to be" verb is: "باش"
and the verbs for this Bon are: می‌باشم، می‌باشی، می‌باشد، ...
But these verb are formal and I personally don't see them used.
So we use a different Bon instead: "هست"
Present verbs: هستم، هستی، هست، ...

Again, this explanation is very short and simple, as I haven't explained regular and irregular Bons. There are also some exceptions that Persian people use unconsciously and don't actually know the rules. And my examples were only of simple verbs.