Hi N2G,
They are not interchangeable.
every day - this is probably the one you mean. Each day, they are a little better than the day before. Something that you do each day (in this case, you improve).
day by day and day to day are closer - each day, you take it as it comes. It deliberately does NOT mean the progression of days, but that each day is an individual event. For example, let's say there is someone who is in the hospital after a very serious car accident; someone asks about his condition and the family says "We take it day by day." That means they experience each day, without looking for what will happen tomorrow or next week. The family could also say "It changes day to day. Some days he seems like he is getting better, but then he will have a setback."
day after day - this expresses a long line of days. It's often used to express something tedious, but not always. Day after day, he went to work, did his job, and came home to eat dinner in front of the TV. There wasn't much joy, but there was no pain either.