An innocent behaviour, like a child's (of qualities, actions etc.,)
We use this almost always in a good sense, with reference to the innocence, meekness etc., of children.
Childlike is usually used of adults, meaning they are "innocent, trusting, or uncomplicated," like children at their best.
An adult behaving like a child, which is inappropriate.
Childish, when used of children, underscores their immaturity and other youthful qualities;
When used of adults, childish is pejorative, meaning "immature, unreasonable, foolish, or fretful." Childishness is frequently cited as an unfortunate symptom of senility.