1) The "verb" that follows the verb phrase "admit to" is a gerund. That means that it is part of a "noun phrase". For example: "He admitted to lying to his parents." "Admitted" is the primary verb, and "lying to his parents" is the "thing/noun" that he admitted. Gerunds, therefore, are verbs that act as nouns in a sentence, and they often have the "-ing" ending. 2) "Agreed on" usually implies that there was an argument/discussion in which two or more people did or did not agree on a fact, or a course of action. "Agreed to" implies that those people decided to DO SOMETHING in the course of their discussion. Example #1: "They could not agree on the interpretation of her testimony." Example #2: "They agreed to move out of their apartment after the lease was up."

Admit (verb) Admitting (progressive tense) Jack admits when he is wrong. Jack is admitting that he is wrong (right now). [Another usage: The patient is admitted into the hospital; The doctor admits the patient into the hospital]. Agree on + noun Agree to + verb The parents agree on two Christmas presents for their kids. The parents agree to purchase (buy) two presents for their kids. Nod (defines agreement) Nod at (an action) Billy gives a nod when he agrees with the teacher. Billy nods at the teacher when he agrees.

1. The pattern is "admit to (something)", so you need a gerund (ing) form instead of a regular verb. The whole trick with gerunds is that any noun can also fit the same pattern. You admit to a crime; you admit to robbing a bank. 2. I guess you mean "agree on (something)" and "agree to (something)". "Agree on" means a compromise; "agree to" means an acceptance. They agreed on a solution. She agreed to the conditions of the contract. 3. The difference in prepositions is the same as "throw at" vs "throw to", or "shout at" vs "shout to". In these examples, "at" indicates a direct target of the action.