The om te plus infinitive construction expresses purpose (in order to) or — after certain verbs and adjectives — simply follows the verb pattern. Purpose example: Ik studeer om te slagen. After adjectives expressing willingness or emotion: Hij is blij om te helpen. After verbs like proberen, vergeten and besluiten: Ik probeer elke dag te oefenen. Note that te sits between om and the infinitive, and in separable verbs it goes between prefix and root: om op te staan.
Zonder te plus infinitive means without doing: Hij vertrok zonder iets te zeggen. Door te plus infinitive means by doing in the sense of means or manner: Door hard te oefenen word je beter. These constructions parallel English without followed by the -ing form and by followed by the -ing form. The subject of the infinitive phrase is always the same as the subject of the main clause — if the subjects differ, a full subordinate clause is needed instead.
The conjunction dat introduces a subordinate clause when the subjects differ: Ik wil dat jij helpt rather than om te helpen, since the subject changes from ik to jij. But when the same subject continues: Ik wil helpen or Ik wil graag helpen. Learning which verbs take te, om te or neither is partly pattern recognition — most verbs of volition take te, while modal verbs take a bare infinitive without te.