-ing forms are used in verb forms with auxiliary verbs, as nouns, and after certain verbs like dislike. They are also used after prepositions. Participles clauses can include -ing forms. Infinitives are used with "to" to express purpose or after certain verbs. They are also used without "to" after modal verbs and verbs like let, make, notice. Gerunds express real, completed actions while infinitives express unreal or future actions, as shown in the examples of forgetting vs forgot and stopping vs stopped. The document provides guidance on the different uses of -ing forms and infinitives.