Hi! I just started learning Esperanto and the same thing that confuses me in both English and Spanish is confusing me with Esperanto.

I've looked up how to find the "direct object" and the "indirect object" and found that a transitive or active verb is always accompanied by a DO. However, I'm not sure if the same rule of asking "what?" to see if a verb is transitive applies to the Esperanto language. How do you all remember how to correctly use the accusative case? Are there any special exceptions I should be aware of?

Also, I began to learn Esperanto because I'm having trouble with learning other languages. I think I am an average learner by all measures. What would I have to do to become reasonably fluent in this tongue? I really want to have something to be proud of.

Thank you if you respond,
eidoloy

eidoloy,

There are two very popular online courses. Here on Lernu "La teorio Nakamura" and on Duolingo. I've completed the later one and doing the first one. I must confess I ought have done the other way. The course in Lernu is in E-o and has a more language neutral approach, while the Duolingo course teaches E-o with English. For a non-native English speaker like me English has caused me way more trouble than E-o. Of course, YMMV.

The basic rule of the verb endings is:

• -i
  - someone or something is something¹ La tablo estas blanka (The table is white) or
  - the action is direct² Mi piedbatis la pilkon (I kicked the ball)
  
• -igi
  someone or something is made to something² Li disigis la batalantojn (He separated the fighters; lit. He made the fighting ones separated)
  
• -iĝi
  - someone or something becomes something³ Ni disiĝis en diversajn flankojn (We got separated into different sides; note: accusative with en denotes movement likewise into does)

¹: The verb denotes a state, nothing is done to anyone or anything, there is no movement nor change of state. blanka/white is a constituent called predicative and predicatives in E-o are in nominative.
²: The verb is transitive, i.e. takes a direct object . Direct objects are always in accusative. In English the accusative is similar to nominative, in E-o accusative has n-ending.
³: The verb is intransitive or non-transitive, i.e. does not take a direct object. In the example above ni/we is the indirect object, the one that changes. In E-o indirect objects are in nominative (as in English).

Now to this basic rule there are many exceptions where i-form denotes a change in state. For instance see my posting elsewhere here in Lernu.

HTH

"Your Mileage May Vary", "Hope That Helps"

I've looked up how to find the "direct object" and the "indirect object" and found that a transitive or active verb is always accompanied by a DO. However, I'm not sure if the same rule of asking "what?" to see if a verb is transitive applies to the Esperanto language.

Mostly that rule applies (if I am understanding what you are saying correctly). However, there are quite a few verbs in English that can be either transitive or intransitive depending on whether a DO is present. Example: "to drown". If I say "Pedro drowned", the word drowned is understood to be intransitive, but with "Pedro drowned a puppy", drowned is understood to be transitive.

It doesn't work quite that way in Esperanto. All Esperanto verbs are inherently either transitive or intransitive. (Though you can change the default transitivity with the use of -ig or iĝ.) Example: droni, meaning "to drown" (in the intransitive sense). Pedro dronis is fine. But *Pedro dronis hundidon is wrong. Can't use a DO with an intransitive verb like droni. It has to be Pedro dronigis hundidon.

The good news is, if you master that, you'll be an expert on direct objects and the accusative case, and that knowledge of how all that works is transferable to learning many other languages.

More good news: Esperanto doesn't use indirect objects, so you don't have to worry about learning a dative case in addition to the accusative case. In English sentences where you would use an IO, instead use the preposition al. For example: I sent Pedro a letter becomes Mi sendis al Pedro leteron (or, Mi sendis leteron al Pedro, or Al Pedro mi sendis leteron).

Metsis: Thank you for the technical explanation. I haven't gotten to the point in any of my courses that mentions those special endings but I'll be sure to keep out for them.

Roch: I mean... That is a complete enough message on its own. That phrase, "your mileage may vary", drives me nuts since it's quite true. I just wish I had a crystal ball that would tell me about my level of suckery.

Trojo: In this link they show what I mean by "what?" --> http://www.grammar-monster.com/glossary/transitive.... And thank you for the clear explanation with the examples. It seems Esperanto is similar to English in many ways. Obviously, though, it does have its differences which I have to keep in mind. Otherwise it's like learning English for a second time, through English, which is just absurd. (However judging by how much I don't understand about my own language, perhaps it's not as silly as it seems.)

eidoloy:a transitive or active verb is always accompanied by a DO

A transitive verb can be accompanied by a DO, but it doesn't need to. The verb "legi" is transitive and can stand without a DO: Mi legas.
However, an intransitive verb can never be accompanied by a DO (except in fringe cases, so called inner or cognate objects).

About your question how one can remember when and where to use the accusative:
The accusative in Esperanto has several usages (DO, duration, direction, measure, etc), but I will stick to DO. The obvious answer is simple: the DO is marked for accusative. However this begs the next question: What is an DO? And this is the really tricky one.

The DO is syntactic function (grammatical role) which is assumed by a phrase. This information doesn't help at all and we arrive at the next question: When does a phrase assume the syntactic function of DO? And this brings as to semantic roles. Semantic roles are easier to grasp as they are more "natural" than grammatical role.

Some common semantic roles (which are of concern here) are:
Agent (AGT): someone of something that actively carries out an action.
Patient (PAT): someone or something that passively suffers an action (often with a change of state).
Theme (THE): someone or something which is directly related to the action (almost the same as PAT, but generally without a change of state)
Experiencer (EXP): someone or something that receive sensatory or emotional input
Stimulus (STI): someone or something that causes sensatory or emotional output

The typical transitive phrase has AGT and PAT:
Mi batas pilkon.
I actively do the kicking, so I am the AGT. The ball suffers from my action (with a change of momentum), so it is PAT.
If you have AGT and PAT, then most probably PAT is a DO.

Another typical transitive phrase has AGT and THE:
Mi legas libron.
I actively do the reading, so I am the AGT. My action is directly related (or directed) to the book, so it is the THE (without change of state, unless books feel uncomfortable when being read. Might be.)
If you have AGT and THE, the most probably THE is a DO.

The third typical transitive phrase has EXP and STI:
Mi vidas lumon.
I am experiencing sensatory input, so I am the EXP. The light is the stimulus, the source of the input, so it is STI.
If you have EXP and STI, then most probably STI is a DO.

But as always, there are exceptions:
Mia patrino ŝatas hundojn.
Al mia patrino plaĉas hundoj.
In both cases "hundoj(n)" is the stimulus STI which causes affection in the experiencer EXP, in this case, my mother. With "ŝatas" STI is the DO as predicted, however with "plaĉi" STI is the subject. So much to predictability.

Some verbs are even instable and allow for various competing constructions (sometimes with a slight semantic shift):
1: mi esperas, ke mi mem min helpos [Z]
2: mi certe helpos al mi ankaŭ sen mono [Z]
Apparently, sometime the one being helped is DO (as in 1), sometimes not (as in 2).

TL;DR

If English is your mother tongue, then just hope that the Esperanto verbs are constructed in the same way as their English counterparts (which is the case in like 95% perhaps), and just copy the pattern, marking the DO with -n. How to spot a DO in English? Replace the phrase you suspect to be a DO with a pronoun. If the pronoun is me, him, her, us, them, but not I, he, she, we, they, then it is most likely a DO (except it is a IO)[1]:
The boys kicked the balls.
With pronouns: They kicked them.
"Them" (= the balls) is the DO, hence La knaboj batis la pilkojn.

----
[1] In which case you will end up with two candidates for the DO. In this case the person is the IO and the thing is the DO. This works almost always.

nornen:
Another typical transitive phrase has AGT and THE:
Mi legas libron.
I actively do the reading, so I am the AGT. My action is directly related (or directed) to the book, so it is the THE (without change of state, unless books feel uncomfortable when being read. Might be.)

This reminds me of those depressed lifts in the The Hitchhiker's Guide to the Galaxy.

nornen:
But as always, there are exceptions:
Mia patrino ŝatas hundojn.
Al mia patrino plaĉas hundoj.
In both cases "hundoj(n)" is the stimulus STI which causes affection in the experiencer EXP, in this case, my mother. With "ŝatas" STI is the DO as predicted, however with "plaĉi" STI is the subject. So much to predictability.

I confess that I have hard time to understand those
  X plaĉas al mi.
  X doloras al mi.
sentences. To me they cause a mental image that someone is ,if not forcing then at least planting, a feeling of niceness resp. pain into me. I feel uncomfortable with that idea. Way more natural for me is therefore to say
  Mi ŝatas X-n.
  Mia X doloras.

Nornen: Thank you for the in-depth explanation. I'll be sure to keep it at hand if I ever need intensive study. And thanks for the tip at the end of your post on how to find the DO.