Let's start with the physical pieces that limit what time is attempting to measure and regulate: the orbit of the earth around the sun, the rotation of the earth, and the cycle of the moon from new to full and back.
None of these things line up; a lunar month is 29.53 days, a solar cycle is 365.25636 days, and a solar day is 24 hours. Worse, these numbers change over time; the day used to be longer, and even today, the length of a day changes almost at random. Not that it would have mattered even 100 years ago; a leap year is all you really need to keep track of time if you're using the sun.
Let's start with the largest useful increment of time: the year. It's a full growing season, shorter than a lifetime but long enough that a lifetime is less than 100 years (generally). Of all the measurements of time, the year has lasted the longest.
The next most useful increment of time is the day; a day and night cycle to mark time passing in a meaningful way. As long as we're on Earth, the year and the day will be immutable. Trying to survive on, say, a strict 25 hour day, will simply not work; the human body is built around the day/night cycle.
Years and days have a fairly solid relationship by the stars, too, so they are easy to keep track of over time.
Between the two are months and weeks:
Weeks are best for delineating work schedules, and organizing days into manageable chunks. These days, the majority of people have a 5-day work week, followed by a two-day break. Soviet Russia tried a 5, 6, and 10 day week, to little success, though that probably has more to do with a rolling calendar than actual number of days worked. There have been a few other experiments with work-week lengths, but as far as I can find, nothing conclusive.
But, what is optimal today with our low-impact and low-energy work wouldn't be best for 100 or 1000 years ago. A 7-day week makes sense for hard, manual work; working sunup to sundown, six days in a row, then taking a seventh day off to rest pushes the body to the limit, but not beyond. That means if you want a week with a different number of days, it should be between 5 and 7 days, but for a long-term solution, 7 days to a week makes decent sense. It allows for the maximum amount of work, without over-stressing the body.
There are three ways to handle months - lunar months, equal months, or no months. Lunar months are designed to line up with the lunar phase; so, each month should be about 29.5 days long, or 29 days on odd months and 30 days on even months. 12 lunar months last 354 days, which means there are 11 or so extra days in a year - and months start sliding around, which means January is in winter one year, but ten years later it's late spring. That's really confusing - don't do that. If you want that system, months shouldn't have names; treat it like we treat weeks today: month 1 is the month at the beginning of the year (and may begin in the middle of a week), through month 13 or 14.
The second option is equal months, not lined up with lunar months. To make a year line up exactly (except for leap years), there should be 5 months of 73 days each. To make weeks line up exactly, there should be 13 months of 28 days each, with 1 day extra (2, for leap years). In a world I created, there are still 12 months, with 8 months having 28 days, and 4 (those with a solstice or equinox) having 35; I eliminated the extra days, but they could well be special holidays or such. A 12-month year is a little easier to divide than a 13-month year. If you want a different-length week, you can divide your months to match.
The third option is to ignore months all together. They don't make a lot of sense if they aren't tethered to the moon, so they aren't much use - just use numbered weeks instead.
You won't be able to avoid time zones. The sun was the best way to measure time until clocks came along, and according to a sundial, noon happens when the sun is high in the sky - and a different time all over the world.
However, time is on your side! Ancient cultures didn't care about time zones; if it took two days to walk from City A to City B, who cares if they are an hour apart, or even six? It wasn't until clocks and high-speed, long-distance travel travel were invented that time zones even mattered at all.
Find a common origin for a year - and to cut down on complaining, make it Year 0. Then, find the number of days in a week: 6 or 7. Then, pick months; I like rolling numbered months, as moon phases matter to many cultures, and it can help delineate time without confusing people too much.
Smaller increments of time honestly don't matter. There's nothing that takes an hour, or a minute, or a second, so go ahead and divide the day into pieces as you want: 24 hours in a day, 60 minutes in an hour, 60 seconds in a minute if you like what we've got; 10 hours in a day, 100 minutes in an hour, and 10 seconds in a minute if you're one of those decimal people; or, 16 hours in a day, 64 minutes in an hour, and 64 seconds in a minute if you like binary.