When do I use Day by day or day to day?

Answer 1

They are quite close in meaning.

"Day by day" refers to process and the passage of time, such as the famous dictum of Emile Coué: "Day by day, in every way, I get better and better". Other examples: "His diary gives a day-by-day account of his life", "Day by day she watched her child grow up", "He's getting stronger day by day".

It has similar meaning to "from day to day", but "day to day" on its own tends to refer to the ordinary things of life, the quotidian. "The day-to-day running of the business is looked after by Mr. X.", "the day-to-day changes in the exchange rate", "She's the one who does the work day to day, I just come in from time to time". There is also the meaning "He lives from day to day", meaning that he doesn't plan ahead.

Note, we usually use hyphens to connect the words when the phrase is an adjective in front of a noun.