What makes our characters look for new beginnings? Are they motivated by a change in year or whatever time period you use in your settings? Is it a question of seeking redemption? (This is always a good story theme, people are fascinated by why a character needs redeeming, I know I am).
Motivations have got to be strong for them to ring true with any reader so we need to be convinced your character is convinced they need redeeming. (Equally we can know the character is faking it but another character, the one they're trying to fool, does not. There, we're fascinated to see whether the first character will get away with it or not). I watched The Naked Truth earlier this evening (starred Terry Thomas and Peter Sellers amongst others). Fantastic film. Absolutely had to see if the blackmailing character was going to be brought down or not. Not a word was out of place in that script. Highly recommend seeing it. (Is easily available online).
If a character is putting a tragedy behind them (or trying to), what do they use to help them make a fresh start? For example, do they pick a particular day on which to make that fresh start and what made them choose it? What support do they have? Do they succeed? (Or is it a question of moving on while accepting they will never be the same person they once were. I would be more convinced by that scenario).
Good luck with your new beginnings - with your characters and with your stories overall. This year should see a new beginning for me as my first flash fiction collection, From Light to Dark and Back Again, is due out (Chapeltown Books), and obviously I'm looking forward to that. But each new story is a new beginning, each new competition entered is a new beginning so there is plenty of scope. The great thing with new beginnings is you don't have to wait until 1st January each year to have them!