Zu, Um-Zu, and Damit

Zu alone is used whenever the first part of your sentence that is the part up to the to , cannot stand for itself.

Example: 

                 (1) Ich gehe ins kino.

                      Ich sehe einem Film.

                       => Ich gehe ins kino, einem Film zu sehen.

                  (2) Ich gehe nach berlin.

                       Ich kaufe die kleidung.

                        => Ich gehe nach berlin, die Kleidung zu kaufen.

Note: Below mentioned verbs are not come with 'Zu',

            - horen - to listen

            - sehen - to watch

            - bleiben - to stay

            - gehen - to go

            - lessen  - to study

    E.g :    zu sehen   X

             

 ' Um - zu '     

when in german, second sentence verb is in first part then that verb come with 'um' and 'zu' as per above rule.

Example: 

               (1)  Ich gehe ins kino, um ruhig bleiben. 

                (2)  Ich nehme das bus, um schnell zu Hause kommen.    

'Damit-so that' 

If the subjects are different in the two parts of the sentence, the final clause is connected to the main clause by the conjunction 'damit' (so that). The conjunction is followed by the subordinate clause (subject + other elements of the clause + predicate at the end). 

Example: 

                  (1) sarkar unterrichtet das Deutsch, damit Mark Peter Prufung bestehen kann.

                  (2) Ich gehe nach mumbai, damit ich meine Eltern besuche

                  (3) Ich nehme das Bus, damit ich schnell zu Hause komme.