Difference between To+verb1st form and Verb+ing form

What is infinitive = To verb 1st form What is gerund = Verb+ing form Gerund = the present participle (-ing) form of the verb, e.g., wandering,singing, dancing, running , laughing Infinitive = to + the base form of the verb, e.g., to play , to write , to sing, to dance, to run. It depends on the main verb which form to be used , infinitive or gerund ? we suggested her to play during the summer . (Infinitive) She called me to cut the ringing phone . (Gerund) Gerunds can be used after certain verbs including enjoy, fancy, discuss, dislike, finish, mind, suggest, recommend, keep, and avoid. 1) After prepositions of place and time. I called her before leaving home. He looked happy after visiting the place. 2) To replace the subject or object of a sentence John likes eating coconut oil. Jumping off a cliff is dangerous, but a real thrill. Infinitives can be used after certain verbs including agree, ask, decide, help, plan, hope, learn, want, would like, and promise. 1) After many adjectives: It is hard to make dinner this late. I find it difficult to describe my feelings about writing research essays. 2) To show purpose: I left for Russia to study Russian. I came to the office to solve the mystery of the missing keys.