Bert (Dick Van Dyke)’s Timeline and Summary

At the very beginning of the movie, Bert acts as a one-man band, entertaining a crowd and amusing people. Then, he predicts Mary's arrival, and guides us (the audience) to the Banks house, setting the scene for the rest of the story.
Later, Mary and the kids meet up with Bert, and jump inside one of his chalk-drawings, which turns into an animated world. Bert chills with Mary, tries to rescue a fox from a fox hunt—generally does his good guy thing.
Next, he goes with Mary and the kids to see Mary's Uncle Albert, who's become stuck floating in the air from laughing too much. He and Bert take to each other, exchanging terrible jokes, until Bert starts floating too. (Eventually, Mary gets them both down).
After Michael and Jane run away from the chaos at the bank, they meet up with Bert and end up having an adventure with him and other chimney sweeps on the rooftops of London. (Mary ends up coming along).
At the end, we see Bert waving to Mary from below as she flies away, problems solved.