OK, so I have been buffing up on my ancient history, and I will just never get used to the way historians do dates — it's such a mess. You have to learn to think backwards if you're looking at dates before the birth of Christ (or even at it, since he was born in the year 4 BC). Then there's the awkward "missing 0 year" that we get from the Romans, because they didn't know how to measure continuous quantities starting at 0. So being born in 4 BC, Christ was 7 years old in 4 AD. This Roman ineptitude also shows up in the way they counted centuries — Christ was born in the First Century BC, and died in the First Century AD, which makes it sound like he was 2 centuries old when he died, when he was actually only 36. (No wonder Roman civilization crumbled.)
The nice thing about the Hebrew calendar is that it starts at , when there just wasn't that much going on. This means that for the overwhelming majority of the periods of interest — modern and ancient — there aren't any negative numbers to futz with. Yet isn't so far back that it would require any additional digits to represent the years. So this is an attractive option. And instead of the Roman system for counting centuries, you just say the 5700s if you're talking about this century, or the 5600s if you're talking about the last one.
The one disadvantage is that the overwhelming majority of dates that have been recorded are in modern times, and which were recorded in the AD system, so all of those dates would change, which would suck. So an alternate proposal is to leave the modern dates the same, but to throw in a "0" year, and then the year before that would be 9999. In other words, make it like an odometer that rolls over at 9999, and we just don't happen to know how many times it's rolled over in the past, because we don't know precisely when the Earth formed, nor do we care to that degree of precision. So Christ would have been born in 9997, not 4 BC. No negative numbers, no missing 0 year, and no changes to modern dates. If you have to do date math spanning the roll-over, you just add 10000 to the later date. So being born in 9997, and dying in the year of 10033 (33 AD + 10000), Christ died at the age of 10033 − 9997 = 36.
While we're at it, we might as well just reform the whole system. (See the Wikipedia article on Calendar Reform for other ideas.)
The New Year should begin at the first midnight after the winter solstice in the northern hemisphere (i.e., December 20~22 in the Gregorian calendar). Sure, each culture has a different religious holiday around that time of year. But that's the point — the only way to be politically correct is to favor none of them (i.e., piss everybody off), and rather, to just go with something secular.
Months of irregular lengths are a problem. The "week number, day of week" is a much better idea, which is already used widely in the manufacturing and shipping industries, where scheduling in a really big issue. The 7-day week is a custom that started back in ancient times, and has been adopted all over the world. In other words, it just works, for whatever reason. So let's go with it. Then, the year simply has 52 weeks, which divides up into 4 quarters of 13 weeks apiece, or likewise into 13 months of 4 weeks apiece (for pretty much the same reason).
If the concept of a "month" is retained, somebody will have to come up with a name for the extra month, since currently we only have 12 names. The International Fixed Calendar proposed by Moses B. Cotsworth in 1902 injected the extra month between June and July, and named it "Sol" in honor of the Sun. I prefer adding the month at the end, and calling it Undecember, which is Latin for the eleventh month (un + dec = 1 + 10 = 11). Of course, it wouldn't be the eleventh month, but for that matter, December wasn't the tenth month (in the Gregorian or Julian calendars), and if people could get used to the 12th month being called December, they can get used to anything.
Regardless, days of the month should start with 0. This way, to calculate the number of days across several months, you just multiply the number of months by the number of days and then subtract, instead of having to get out a calendar and count each day with your finger. (Rome fell for a reason.) So here's what a month would look like (i.e., any month, since they'd all be the same):
52 weeks x 7 days = 364 days per year, which of course is 1 day shy of a normal year, and 2 days shy of a leap year. But rather than having the weekly schedule change from year to year, I prefer the idea of absorbing the discrepancy once a year, with 1 festival day at the end of each normal year, and 2 at the end of leap years. This way, the year will always start on the first day of the first week, and proceed as it always does, making it easy to plan things from one year to the next. (This would make it a perennial calendar, beginning every year on the same weekday, which was first proposed by the Rev. Hugh Jones in 1745.)
Since continuous quantities (such as time) are properly measured starting at 0, the first day of the week (i.e., Sunday) should be day #0. Likewise the week numbers would start at 0. So the first day of the year should be 00-0. This would make the last regularly-scheduled day of the last week 51-6, the first festival day 51-7, and the second (on leap years) 51-8.
Then we just have to come up with a standard way of writing the numbers that is not ridiculously ambiguous, the way our various current systems are. Hence today is 13-08-12, or 08-12-13, or 12-08-13, depending on which style you prefer. This would be 2013-32-1 in the "ywd" (i.e., the year-week-day system), and you'd actually be able to tell what day it is just by reading the number — no guessing necessary!