Allow me to simultaneously help and further confuse you. >:)

As other users have pointed out, the main use for the accusative case in Esperanto is the direct object. English users are actually used to using a similar form with pronouns (though this is actually the oblique case). While English speakers are likely to confuse who and whom, there is almost always a clear distinction (except when listing them with conjunctions, "me and John went to the store" is a common mistake).

So one thing that may help is to replace nouns with a pronoun. Before translating "I hit John", change it to "I hit him", and you'll know because the word "him" was used instead of "he", that the Esperanto translation must be "Mi frapis Johanon." Remember that "it" and "you" don't change in the English oblique, so stick with "I/me", "we/us", "he/him", "she/her (but not the possesive her)", or "they/them".

However, there are caveats that must be taken because the oblique case doesn't perfectly overlap with the accusative case. "She came with him" uses the oblique, but because it is the object of a preposition, and those are almost always (see below for the exceptions) nominative in Esperanto, so "ŝi venis kun li." "I gave him the book" uses the oblique, but instead of serving as the direct object, it serves as an indirect object! This must be turned around before you translate it, as Esperanto doesn't have ditransitive verbs, so if I make it look like the previous example, it becomes "I gave the book to him", and thus "Mi donis la libron al li." This one can crop up if you use the passive method mentioned in another comment: "He was given a book by me" is a legal passive periphrastic construction in English even though "he" was the indirect object in the original active sentence.

Don't forget, also, that English uses the oblique in predicates, even in the verb is a copula! "It was him" is not accusative in Esperanto because copulae like "esti", "iĝi", etc. don't take direct objects, and the noun on the other side of the verb is technically a predicate nominative. One more tricky feature is when there is a secondary predicate. "He paints the door red" is not "Li farbas la pordon ruĝan" because due to adjective agreement, that makes it mean "He paints the red door." Instead, you must realize that there's an implicit extra clause in there: "He paints the door to be red", so the translation is "Li farbas la pordon esti ruĝa", or more commonly "Li farbas la pordon ruĝa".

Another fascinating (to me anyway) use of the accusative is how it is used in relative clauses. "Tio estas libro, kiu plaĉas al mi" is pretty simple, right? "Here's the book that I like". No accusatives. But what if we use ŝati which is inverted from plaĉas? "Tio estas libro, kiun ŝatas mi." or "...kiun mi ŝatas". Even though kiun is, in a way, modifying libro, they don't agree in case. The case of kiu must be determined by the verb in its clause and in this case kiu indicates the direct object of ŝatas. While the accusative normally affords Esperanto a fairly free syntax, in this sentence it is restricted to object-verb-subject or object-subject-verb. If you're one of the increasingly rare anglophones that can properly use whom, this skill can come in use here.

Of course, all this direct object stuff will become second nature, and you won't have to use tricks like these once you get used to it -- though, I admit, that I still forget the -n occasionally. The tricky part that they don't teach you at the beginning, though, is that Esperanto doesn't use the accusative case exclusively for direct objects!

One bonus use of this case is using the accusative for motion. "Kien vi iras?" "Mi iras hejmen." "Where are you going?" "I'm going home." These are adverbs that describe direction and must be in the accusative. "Mi metis la libron sur la tablon." "I put it on the table." Even though tablo is an object of a preposition, the book wasn't on the table before, so you have to use the accusative to show its journey to the table. If you can use a "-to" preposition like "into" or "onto" (but not "to" by itself), then use the accusative. I know it's weird, but if the Ancient Romans could get used to using the accusative like this, then you can too.

The other use that I can think of (maybe the more veteran Esperantists out there can correct me if there are more or if I'm overgeneralizing) is when the noun would come after a preposition, but it got removed. This one is especially bizarre and hand-wavy to me, so I'm still getting used to it. The most common example is with time expressions like this one from lernu's page on this subject: "Unu tagon estis forta pluvo. = En unu tago...", but apparently this can extend as far as something like "Mi ridas je lia naiveco. = Mi ridas pro lia naiveco. = Mi ridas lian naivecon." Crazy, right?

Whoever said Esperanto was easy? :)