Yes, you're right - "die Zeitung" is a direct object in this sentence and it's accusative. Except after a handful of special verbs (e.g. helfen, danken), direct objects always take the accusative case.
Only the articles of masculine nouns differ in the nominative and the accusative cases.
Masculine nouns: der (nominative) - den (accusative)
Feminine nouns: die (nominative) - die (accusative)
Neuter nouns: das (nominative) - das (accusative)
Plural (all genders): die (nominative) - die (accusative)
See also this chart: http://de.wiktionary.org/wiki/der#Artikel
Normally the direct object takes the accusative case. However, after a limited number of special "dative verbs" such as "helfen" (to help) or "danken" (to thank), the direct object takes the dative case. E.g. "Ich helfe dem Mann" = I help the man. "Dem Mann" is dative masculine. "Ich danke der Frau" = I thank the woman. "Der Frau" is dative feminine.
Definite article (= the)
Masculine nouns: der (nominative) - den (accusative) - dem (dative)
Feminine nouns: die (nominative) - die (accusative) - der (dative)
Neuter nouns: das (nominative) - das (accusative) - dem (dative)
Plural (all genders): die (nominative) - die (accusative) - den (dative)*
*Note that in the dative plural, not only the article changes, but also the noun itself. It adds an -(e)n. E.g. "Die Männer (nominative plural) sind groß.", but: "Ich helfe den Männern" (dative plural). The only nouns that don't add an -(e)n in the dative plural are those whose nominative plural already ends in -n (e.g. die Frauen) or ends in -s (e.g. die Autos).
List of special dative verbs: http://german.about.com/library/verbs/blverb_dativ.htm
PS: You could try to memorize the definite article in the following way: "Reesee, Neesee, Merman", i.e. RESE (nominative), NESE (accusative), MRMN (dative). These are the last letters of the articles for masculine, feminine, neuter and plural in the respective case, see the list above.