Pope Gregory XIII was a doofus as was Julius Caesar. Did they not foresee the importance of a logical calendar in a technological society? Here are some changes I would make if given supreme dictatorial power over the Free World.
1) New calendar starts the day after the winter solstice (i.e. the winter solstice is the last day of the year). The days start off short, get long, and once they're short again we start a new year. I could compromise by going with the summer solstice or one of the equinoxes, but the current system of "ehhh, I guess we'll start the new year like a week and a half after the solstice" is stupid and dumb and I hate it.
2) 12 months, 30 days each. Months 1, 3, 6, 9, and 12 all get an extra day. That adds up to 365.
3) No leap years. The final day of the year will be 0.2422 days longer to help the rotation and the revolution counts catch up to each other.
4) Better month names. September, October, November, and December are months 9, 10, 11, and 12 just so that we could fit in two holidays named after emperors who have been dead for like 2000 years? That's insane. Nope, scrap the whole system. Pick some new naming scheme. Name them after stars, rocks, trees, historical figures, elements, or dinosaurs. The current system has no cohesion or logic.
5) Now that there's no shift in which day falls on which date, all holidays are static and better spaced out. Every month gets a holiday and they all happen either on a Monday or a Friday. None of this "oh, sometimes it lands on a Wednesday". That's not a lottery I'm interested in gambling at. Now I realize many religious and cultural celebrations may follow a lunar calendar or that certain groups may be attached to celebrations that are already tied to specific days. Well tough tamales! You should've thought of that before electing me supreme leader of the world.
I think that's a pretty good start, and it's pretty easy to fit a 5 step plan into my manifesto. I could go into detail on my plans for a 3 day weekend, second Christmas, or regulations about which holidays warrant fireworks and which ones do not, but I think I can save that for the supreme world leader election circuit.