There are many uses for the infinitive in German, sometimes with zu, sometimes without it. To keep it simple, I sum up the three cases with zu for someone who knows English:
First, sometimes when English uses its gerund, which German lacks:
Zu lesen fällt mir leicht.
Reading comes easy to me.
BUT as German speakers adore nouns, the alternative with a noun made from an infinitive is far more common. (Note the caps on Lesen.)
(Das) Lesen fällt mir leicht.
Second, anywhere English uses its to-infinitive:
Sie gab mir eine deutsche Zeitung zu lesen.
She gave me a German newspaper to read.
Third, in a special clause: the Infinitivsatz. That clause is a very common replacement for an Objektsatz if its subject is identical to that of the main clause.
Sie verbrachte den Nachmittag damit, dass Sie die Zeitung las. (Objektsatz)
Sie verbrachte den Nachmittag damit, die Zeitung zu lesen. (Infinitivsatz)
She spent the afternoon reading the newspaper.
If you use the Objektsatz instead of the Infinitivsatz, German speakers would automatically assume the subject has changed and are puzzled if it hasn't.
This third case is uncommon to English speakers but you can easily spot it because it's used in a lot of cases where English uses its to-infinitive or its gerund. See the other two cases, remember you have to place a comma, and –unlike English– it's a clause of its own.