Do these three words mean similar things in this sentence? English question?

Although the three seem similair, the one with the closest meaning to what you are trying to convey is childish simply because it is the one most commmonly used. Infantile is close in meaning to childish but is usually reserved for more specialized things that infants would do. Childlike has the connotation to usually mean the characteristics of a child. Childish is more connotated with insult and describing immaturity. The fact is though you wouldn't be technically wrong for using any of the three.