[subject] + 一 + [time word] + 比 + 一 + [time word] + [adjective] + (了)
一 + [measure word] + 比 + 一 + [measure word] + [adjective]
一天比一天 is a 比 comparison folded back on itself: each day is more so than the one before. The same time word appears on both sides of 比, and the adjective comes last. A final 了 marks the change as already under way.
With 个, 件 or another measure word instead of a time word, it compares the members of a group: each one more so than the last, so all of them very.
As in any 比 sentence, the adjective takes no 很 or 非常.
越来越 also means "more and more", and the two don't stack: pick one.
一天比一天 sounds measured, change visible at each step; 越来越 is the general word.