First, some preliminaries that some others have mentioned: I would keep the basic system, since it seems to work fairly well, and there are very few calendars that don’t follow them. So, we will need days, weeks, months, and years.
Going from biggest to smallest:
The epoch or year 0, and there is an explicit year 0 in this calendar, is the first day of the year 4713 BC. This is the beginning of the Julian date system, which is convenient since it precedes almost all human history. That would make this year 6723. (I like this concept, but I am not married to this particular epoch. Other good ones will help.)
As for getting the calendar to stay in synch with the seasons, you have three options. Leap days, leap weeks, and leap months. A month seems too long, and would make scheduling recurring annual things, like school calendars and the like a real pain. Weeks would be okay, but they allow a rather wide variation from the solstices and the equinoxes. So, days it is. The leap day would be on the last day of the year, since that is the easiest place to put it. There are a number of ways of determining where to put the leap days. I would put the anchor point of the calendar at the vernal equinox of the northern hemisphere. It just so happens that that point is the most stable (although it won’t necessarily always be) of the bunch. Conveniently, this means we can riff off the Gregorian calendar which also uses the vernal equinox as its anchor. As for leap years, the Gregorian calendar has a somewhat suboptimal system, but one that is fairly easy to remember. The other way is to have 8 leap years every 33 years. (What you do is divide the year number by 33, and if you get a remainder that is divisible by 4, except for remainders of 0, that year is a leap year.) I kind of like this system, so I will go with it. But instead of having the anchor point of the calendar, the vernal equinox, fall on March 21, as it does now, I would have the vernal equinox be close to the beginning of the year.
There are calendars with thirteen months, but it seems that commerce prefers that the year can be broken down into quarters. Now, while this sort of division has arguably led to shortsighted decision making, let’s go with it. So we need either 8 or 12 or 16 months. 12 is probably the best since it divides 3 and 4, but just to be different, let’s go with 16. This gives us a smidge less than 23 days a month. I would partition the days integrally into months as evenly as possible. This means that months 1–7 would have 23 days, month 8 would get 22, months 9–15 get 23, and month 16 gets 21 (or 22 in leap years).
I would divide the year into weeks of 8 days. This either leads to a cyclical pattern, where every year has each day of each month change the day of the week it falls on (the situation now), or you could decree that the week starts over each year. I know that many calendar reforms try to have the same date on the same day of the week each year, but I’m not sure that’s all that great a feature. Some people’s birthdays would never be on the weekend for example. And (and this really gets me since it happened a lot although not every year) some people’s birthday would always be the first day of school. Is this enough to justify avoidance of the interruption of the cycle of days of the week? Perhaps. Probably not. But I will do it anyway.
As for days, I would keep the 24 hours of 60 minutes of 60 seconds, since attempts to change this (at least in the West) have not gone well at all. But I would disassociate the second that determines things in the metric system from the second that governs clocks. The clock second would be increased to avoid or minimize the leap seconds that get added from time to time. I imagine a situation where the clock second is redefined every 50 years to keep clocks in synch with the rotation of the earth. Doing this, however, may prove to be a pain in the ass in practice. Days begin at midnight, and end the instant before.
Past that, we need names for stuff. Each language can do whatever it wants, but I would suggest we keep things roughly the same, looking through history to fill in the gaps. For example, there used to be months named Quintilis and Sextilis (perhaps anglicized to some degree), so we could run with that. Perhaps we could borrow from other calendars as well, such as the first months of the Hebrew calendar (Nisan) and the Islamic calendar (Muharram). To keep January from falling in the spring, let’s do a bit of a shift, and start the year in April: We might get a list of months that looks like this: April, May, Nisan, June, July, August, Sextilis, September, October, Muharram, November, December, January, Quintilis, February, and March.
As for the days of the week, they are currently based on the planets as they were once defined. We could either go with that same idea with the current definition of the planets (giving Mercuryday, Venusday, Earthday, and so on), or we could just add Uranus or perhaps avoid the jokes that go along with that and use Neptune. I would go with Neptune, and I would stick Nepday after Wednesday. This gives the days of the week as: Sunday, Monday, Tuesday, Wednesday, Nepday, Thursday, Friday, and Saturday. I will use the system where the week begins on Sunday.
I have no idea what all this makes today’s date. The math is really, really, error prone, and I just gave up.