Despite the rain, the picnic continued as planned. The comma is needed here to separate the introductory clause "Despite the rain" from the main clause, clarifying the relationship between the ideas.
The cake, which was baked fresh that morning, was a huge success. The commas are necessary to set off the non-restrictive clause "which was baked fresh that morning," indicating that this information is additional and not essential to the main point of the sentence.
I went to the store, the bank, and the post office. The comma before "and" is part of a list, ensuring clarity by separating each item and preventing any confusion about the number of items involved.
She said, "I'll be there in five minutes." The comma is required to separate the introductory phrase "She said" from the quoted speech, making it clear that the following words are a direct quotation.