Also bear in mind, that "devus fari" doesn't mean "should" do, but "would need to do".

Correct usage:
If I wanted to buy a car, I would need to borrow money.
Se mi volus aĉeti aŭton, mi devus (bezonus) prunti monon.

Wrong usage:
He shouldn't do that.
Li ne devus fari tion.

Here you need the -u form:
Li ne faru tion.

I told him he should go and speak to his mother.
Mi diris al li, ke li iru kaj parolu kun sia patrino.

So I was just taking a look at this article and it was summing up for me that:

devus = should
volus = would
povus = could

So in that case are these sentences correct?

Vere mi devus aĉeti la aŭton.

Mi volus iri al Tokio, se mi havus la monon.

Fakte mi povus venki tiun boksiston en konkurso.

Metsis:PrimeMinisterK,

This is an advanced topic, with which you should have waited for later.

Sudanglo pravas pri tiuj esprimoj, kiujn mi pensas.

Let's take an example: "I hadn't done the error, if you had told me the truth". If you use la imaga modo to translate that into Esperanto, then the Esperanto sentence means that I made an error because you didn't tell me. There are (at least) three ways to express (1) the time and (2) the imaginarity.
 

• erroneous: Mi ne farintus la eraron, se vi dirintus al mi la veron
  
• too complex: Mi ne estus farinta la eraron, se vi estus dirinta al mi la veron.
  
• best: Mi ne farus la eraron, se vi antaŭe dirus al mi la veron.

Furthermore such forms as devintus, povintus for "should" and "could" are outright wrong.

I think you're right, this is too advanced for me right now. But I appreciate the explanation.

nornen:Also bear in mind, that "devus fari" doesn't mean "should" do, but "would need to do".

Correct usage:
If I wanted to buy a car, I would need to borrow money.
Se mi volus aĉeti aŭton, mi devus (bezonus) prunti monon.

Wrong usage:
He shouldn't do that.
Li ne devus fari tion.

Here you need the -u form:
Li ne faru tion.

I told him he should go and speak to his mother.
Mi diris al li, ke li iru kaj parolu kun sia patrino.

Hmm.

So this introduces another thing that is confusing me: My understanding is that the -u ending is used when issuing commands, but that doesn't seem to be exclusively true.

For instance, your example here (Li ne faru tion.) is not a command at all. You are not even speaking to him. You are merely issuing an opinion to some third-party.

How do you handle "would have" in Esperanto. Would you use the conditional for this, or something else?

For instance, "I would have gone to the party, but my ex-girlfriend was there."

Mi irus al la festo, sed mia eksino estis tie.

Volus = would like, would want

Vere mi aĉetu la aŭton. I should buy the car.

Mi irus al Tokio, se mi havus la monon. I would go to Tokyo

Fakte mi povus venki tiun boksiston en konkurso. Actually I could beat this boxer

nornen:
Vere mi aĉetu la aŭton. I should buy the car.

Hmm. So in the article this example is given:

Mi devus aĉeti melon = I should buy a badger.

So did the article writer get this one wrong? Wouldn't "vere mi aĉetu la aŭton" be "I WOULD buy the car" and not "I SHOULD buy the car"? If not, then how do you say "I would buy the car"?

nornen:Mi irus al Tokio, se mi havus la monon. I would go to Tokyo

Okay, that makes sense.

nornen:Fakte mi povus venki tiun boksiston en konkurso. Actually I could beat this boxer

So that's a little different from what I was trying to say. I mean to say, "I WOULD beat this boxer." That is, I would beat him if he would fight me.

Would you say, "Fakte mi venkus tiun boksiston en konkurso"?

This is most certainly going to confuse you a lot…

PrimeMinisterK:

nornen:
Vere mi aĉetu la aŭton. I should buy the car.

Hmm. So in the article this example is given:

Mi devus aĉeti melon = I should buy a badger.

So did the article writer get this one wrong? Wouldn't "vere mi aĉetu la aŭton" be "I WOULD buy the car" and not "I SHOULD buy the car"? If not, then how do you say "I would buy the car"?

Firstly stop translating word by word, translate the idea. Secondly there is a reason, why PMEG uses own terms like la imaga modo or la us-modo instead of the conditional mood. With a term like "the conditional mood" you carry the meaning and usage from your native language into Esperanto and that might not be correct.

The linked article seems to explain the chapter in PMEG to English speakers. Since English is not my native language, I (how to put this diplomatically) have a lesser interest to dive into all nuances in it. Nevertheless the writer Andy explains the two meanings of "I should buy a badger", and these get translated to as Nornen hinted:

• Mi aĉetu melon.
  
• Mi devus aĉeti melon.

PrimeMinisterK:

nornen:Fakte mi povus venki tiun boksiston en konkurso. Actually I could beat this boxer

So that's a little different from what I was trying to say. I mean to say, "I WOULD beat this boxer." That is, I would beat him if he would fight me.

Would you say, "Fakte mi venkus tiun boksiston en konkurso"?

Think about the meaning of "I would beat this boxer". You yourself said "I would beat him if he would fight me". Clearly you are talking about some future event, and naturally future is uncertain.
 

• Fakte mi povus venki tiun boksiston en konkurso : Actually I could beat this boxer in a match.
  
• Fakte mi povos venki tiun boksiston en konkurso : Actually I would beat this boxer in a match.

To summarize

• la imaga modo (the us-ending) expresses imaginarity
  
• la os-finaĵo expresses uncertainty in future

Metsis:
Firstly stop translating word by word, translate the idea. Secondly there is a reason, why PMEG uses own terms like la imaga modo or la us-modo instead of the conditional mood. With a term like "the conditional mood" you carry the meaning and usage from your native language into Esperanto and that might not be correct.

The linked article seems to explain the chapter in PMEG to English speakers. Since English is not my native language, I (how to put this diplomatically) have a lesser interest to dive into all nuances in it. Nevertheless the writer Andy explains the two meanings of "I must buy a badger", and these get translated to as Nornen hinted:

• Mi aĉetu melon.
  
• Mi devus aĉeti melon.

Not "I must buy a badger" but "I should buy a badger."

And what I am saying is that if "My devus aĉeti melon" means "I should buy a badger," then shouldn't "Vere mi devus aĉeti la aŭton" mean "I really should buy the car"?

That's what I would think, but Nornen seems to be correcting me on this.

Metsis:Think about the meaning of "I would beat this boxer". You yourself said "I would beat him if he would fight me". Clearly you are talking about some future event, and naturally future is uncertain.
 

• Fakte mi povus venki tiun boksiston en konkurso : Actually I could beat this boxer in a match.
  
• Fakte mi povos venki tiun boksiston en konkurso : Actually I would beat this boxer in a match.

To summarize

• la imaga modo (the us-ending) expresses imaginarity
  
• la os-finaĵo expresses uncertainty in future

Now hold up.

First, imagine a scenario where you are merely a spectator sitting in the crowd at a boxing match and you say, "I could beat this boxer" or "I would beat this boxer." You are saying two different things. One expresses doubt about the outcome, while the other expresses supreme confidence. In the first sentence you are merely expressing that victory would be a possibility, while in the second you are expressing your belief that victory would be assured. I'll also mention that this would fall into the imaginary category, because the speaker knows that in reality he is never going to fight that guy. It's a hypothetical scenario.

Furthermore, if you say, "Fakte mi povos venki tiun boksiston en konkurso," then I would think of that as "In fact, I WILL beat this boxer in a match." To me, that means that a match has been scheduled and the person speaking is certain he will win. But if you say, "I WOULD beat this boxer in a match," then you're saying you would beat him if a match were to actually occur, but as of yet no such match has been scheduled. Are you then saying that you would use "povos" in either situation and context has to determine the meaning?

Ĉi tiu mesaĝaro estas tia, kia povus plorigi grizharulon.

La samideanoj gaje enbataliĝas, ne zorgante pri kripligo de la angla, trudante teoriojn pri la Esperanta gramatiko, ignorante kiel la lingvo estas fakte uzata kaj senĝene konfuzante la iniciatinto de la diskuto.

PM-K, the Tekstaro is a corpus created for language research (millions of words from the very earliest writings to recent usage) which allows you to check current and historical usage in Esperanto.

How happily a certain party condemns 'povintus' as wrong despite 180 hits in the Tekstaro from a variety of authors over a period of 70 years) whilst frequently using that abomination of a pronoun 'ri', 4 hits all since 2012. Personal preference is not a sound basis for objective language description.

How dogmatically, the theory is advanced that -us is completely timeless, whereas careful examination of the actual usage shows this to be misleading. [There is no way that Ĉu vi ŝatus ion trinki can mean would you have liked something to drink].

I'll say nothing about the misuse of English in this thread. If you are not a native speaker this is forgivable.

PM-K, If you want to know the truth about devus, povus and volus, get to know how to search the Tekstaro, and then think how you would translate the usages you find into English. Then cross-reference with the explanations in PMEG, or look up the examples in PIV.

If you want to look at -us with other verbs you will need to enter the search as \w+us (backslash w + us)