If you don't care about German grammar, skip this post. :P
I had learned in school that when a transitive verb has just a direct object, that object was always marked in the accusative case: "I see him." And if a transitive verb had two objects (a direct and an indirect object), then the former was accusative and the latter dative: "I gave him the book" or I gave the book to him." Wikipedia agrees: "In Latin and many other languages, the direct object is marked by the accusative case, while the indirect object is typically marked by the dative case."
However, it does not seem to be that simple, at least in German. (You saw that weasel-word "typically" above, right?) Wikipedia also says, "The dative is generally used to mark the indirect object of a German sentence. ... Some German verbs require the dative for their direct objects. Common examples include folgen, helfen and antworten."
So it looks like my understanding of accusative/dative for direct/indirect objects is correct generally, but in German there are significant exceptions. I suspect that these exceptional verbs that require their direct object to take the dative will have to memorized, just like whether a verb is strong or stem-changing must be memorizing for each individual verb. There are even lists for this.
@DoubleTranslate's weekday exercises are really great for German beginners to get some practice in. I highly recommend following them, especially if you're self-taught and don't otherwise have homework assignments.