Infer means to come to a conclusion based on facts or observations; to conclude from certain evidence (literally, to bring in, to deduce): “The police could infer that the thief entered the house through the backdoor.”
Imply means to suggest a consequence rather than to make a clear statement: “I imply from your cold shoulder treatment that I am an uninvited guest.”
Insinuate has the following meanings:-
1. To insert something into a context with an indirect suggestion: “The story insinuates that greed leads to corruption which further leads to many evils.”
2. To gradually make oneself a part of a group, a person’s life, etc.; often by behaving in a dishonest way: “He has been able to insinuate himself into the inner circle of the minister.”