You're describing a 1:2:6:12:12:36:52 "Laplace resonance" There are lovely systems with orbital resonances in real life.
The calendar you need must be based on the least common multiple of these numbers. So we factorize each one: -;2;2x3;2x2x3;2x2x3;2x2x3x3;2x2x13. Now we string together enough factors that each one has enough pieces to work with, which is to say 2x2x3x3x13. So our LCM is 36x13 (we could have seen everything but 52 will evenly divide 36). That means the calendar is 468 periods long, where the period is half a day. So it's a 234 day calendar. Makes sense to split it into 13 months of 234/13 = 8 days, no wait, make that 8 months of 13 days. Each month starts with the 26-day moon getting to one end or the other of its orbit. The daily and twice-daily moons don't need a calendar.
The 3, 6, and 18 day moons don't fit the calendar neatly, so so need either a week or a "fortnight" of 18 days, call it an "eightnight" (the apostrophe for 'teen' in there isn't written, I dunno why). Okay, so the eightnight is longer than a month, that's a trouble. Maybe have a thirtnight and an eightnight and not say which is the week and which is the month, or give them different names; you could almost use an Earthly precedent.
When to start it depends on conjunctions - what's the most interesting? With a 3, 6, and 18 day cycle, the three moons could line up at once, or the 6 and 18 might line up and every time the 3 is skewed to the side. Similarly with any of the others. But once you have a most interesting day, the thirtnight and eightnight counts start from that, with the whole thing repeating in 234 days.