I have about five main planets in my fictional universe and like any planet, their days, months, and years are all different. Are there any problems that could mess the narrative up, since my characters move from planet to planet usually every new story arc? Problems with timing and discussing how many days they've been on certain worlds?
A bit of another rough question that's hard to explain, I'll try to edit the question again if I have to.
Since there are so many habitable planets and travel between these, the societies will have already developed some sort of standard calendar so that the people on different worlds can have an accurate reckoning of time when communicating with each other.
Most likely this would take the form of a dominant planet imposing its "Standard Time" as the basis for the entire solar system, and then having the rest of the planets calculate their own calendar date and time based on that. Similar to our time zones and Coordinated Universal / Greenwich Mean time.
Stay with the standard of the planet you are talking about
When you travel from California to Paris, and start interacting with the people in Paris, eating in Paris and going to the movies in Paris, you don´t use any more the California timezone. You don´t say "The movie will start at 4:30 am California time", you stick to the timezone of the place you are currently interacting with.
And when you are in Paris, and you talk about the place you came from (California), you say "I usually wake up at 6:00 am in California". You don´t say "I usually wake up at 3:00 pm Paris time". Neither "I usually wake up at 6:00 hours GMT-7"
Also, in some countries you buy gasoline in gallons, and in some others you buy gasoline in liters. Differences in quantities and measures exist here on Earth already, and we try to adapt to the location we currently are, and only do the conversions when need to use a reference.
So, I think it would be better if your characters mantain the standard of the planet they are in. For example, when they are in Mars and they say "We have been traveling for 15 days" they are talking about Martian days. And if they need to reference another planet, they use then the calendar of the other planet: "Within five Earth days, my son will celebrate his 25 th birthday in Wisconsin."
You have a problem, but what you've described isn't it
Your issue only exists if all space travel is seriously sub-light speed (non-relativistic). Your calendaring problem has problems in spades with relativistic flight.
This is because the meaning of time changes as you approach the speed of light. How much time passes for you, the traveler, measured to the second, would depend on knowing exactly to quite a few decimal points the exact speed you're travelling second-by-second.
which is a fancy way of saying, even if you set your clock to Altarian time when you left Altair VI, by the time you reach the Degobah system, your time reference is hosed.
Coordinating planetary time is trivial — if you have a good starting point
Let's say you have a magic portal such that someone on the Altair side could call out and say, "It's five o'clock right... about.... NOW!" and someone on the Degobah side syncs their clock. After that it's pretty straightforward math to convert back and forth between the two time systems.
It's that pesky relativistic issue that's the problem. The math exists... but you actually need incredibly precise velocity/acceleration control for that math to even have a chance.
So... in the long run, having different orbital and rotational times isn't the big problem. It's syncing up your travel time that's the real problem.
For a solar system spanning polity, all the settlements wherever they are could use UNIX time.
Unix time (also known as POSIX time or Epoch time) is a system for describing instants in time, defined as the number of seconds that have elapsed since 00:00:00 Coordinated Universal Time (UTC), Thursday, 1 January 1970,1[note 1] not counting leap seconds.1[2][note 2] It is used widely in Unix-like and many other operating systems and file formats. Because it does not handle leap seconds, it is neither a linear representation of time nor a true representation of UTC.[note 3] Unix time may be checked on most Unix systems by typing date +%s on the command line.
Unix time has the advantage that anything could be timestamped anywhere in the Solar system and there is an unambiguous record everywhere as to when the event occurred. For example, do you remember where you were at 1000212400000? Unambiguous timings also has advantages in doing banking, securing contracts and coordinating spacecraft flights between the ATC zones surrounding each planet, moon and space station.
Since UNIX time also has a very well known start date 1 January 1970 00:00:00 UTC, so all events can even be coordinated between events on Earth as well (assuming sub relativistic spacecraft to other star systems keep their clocks running and in order throughout the flight).
So UNIX time should be the preferred method of time and calendar keeping in a Solar System.
One problem you will have is that the calendar from one world will not apply to any other world. Mars takes longer to make a full orbit around the sun than the Earth does. So Earth's winter will happen at different times of the Martian year. So the Earth calendar doesn't actually carry any information for what its like on Mars. If you're writing a hard-science work, it could be easy to lose track of what season it is on each planet.
So, I'd follow up on some lesser-known Russian Sci-Fi (again!).
The book is quite unpopular and admittedly quite badly written, but is a hallmark of many nice engineering and worldbuilding solutions.
Half of our physics (in its usual form) and almost all of the engineering is based on keeping the local second same as the SI second. You just don't want to mess with this.
You won't have good and fitting "Earth" hours in a day, so quasi a "Martian" minute would be like 100 seconds, which is confusing.
Even worse, you won't have "Earth" months and years. So, loan from everywhere, from French revolution to fantasy books.
So, each "world", each planet gets its own, unique time units from minutes to years, ideally with unique names, in order to not confuse the innocent.
You need a time scale larger than a planet, perhaps then a galaxy instead?
The issue you have is each planet has its own time frame, its own orbital speed and year, and its own seasons, and perhaps moon, no moon, or dozens of moons (months therefore make no sense).
So you need an external reference point to measure time - for instance, the orbital period of a special star system around the galaxy. Using this, all planets can synchronise to the same reference. By subdividing the orbital period of the star system, it may be possible to create decimalised dates instead of 'Months' and 'Years'.
If the star system is a pulsar, this may be further used to determine seconds (or milliseconds).
This may even be useable in a relativistic sense - compensating for your speed the referenced star system (or pulsar) would still be a measurable unit to determine time, and landing at a target foreign star system the reference will line up again.
Every Planet Has Its Own Time
In your question you mention "days, months, and years" but really each planet has just two fundamental descriptors for its own time. Days and years are prescribed by the planet's orbit and rotation. All other units of time (seconds, months, decades) are just useful subdividers/multipliers for these two fundamental units.
The time it takes to orbit the star is the year, which is useful for knowing the progress of seasons. This is more important on a planet like Earth with pronounced axial tilt, which leads to significant insolation differences throughout the year. In comparison, on a planet with no axial tilt and a roughly circular orbit, keeping track of years would be fairly unimportant.
The time it takes to rotate is a day, which is useful for knowing the light/dark cycle. This will affect weather, the biosphere, your characters circadian rhythms, etc...
As far as I know, there is no rule for how a planet's speed of rotation should compare to its orbital period. For example, a day on Venus is longer than a year. So the length of a year on your planets should correspond to their distance from the star (further=longer year https://en.wikipedia.org/wiki/Orbital_period), but you can pretty much decide the length of the day on each planet.
I should note that there are finer aspects of orbital mechanics that can alter these fundamental timekeeping units. For instance, apsidal precession shifts an eccentric orbit so eventually seasons occur at different times in the year. The length of day can change in some situations (Earth days are getting longer as the moon moves further away). These changes take really long times compared to human lifespans and shouldn't matter unless your narrative spans that kind of time.
If your characters are in a developed system with interplanetary trade/government/diplomacy, they will probably have a unified time system. This doesn't mean each planet wont have their own calendars/time to keep track of where they are in a day or year, but they will need to have a shared system as well that they can reference with each other. A shared system would presumably go by the most powerful planet's time, or maybe by some time system in the reference frame of the star itself.
If your characters are just hopping around an undeveloped system with no real ties between the planets, then they won't bother much trying to tie their time to a planet they have previously visited. If they have to keep a fairly normal sleep cycle (e.g. 8 hours per 24 hours) and the length of a day on their current planet is much longer or shorter than they are used to they probably wont be awake with the sunrise and asleep with the sunset.
I'm assuming you were not worried about relativity. In reality, if one planet kept the official unified time, the other planets would have to make a relativistic adjustment to their devices for keeping the unified time. This is because the planets would be moving different speeds and thus their perception of how fast time moves would be different. These effects would matter very little for most people, but I guess if your system has existed for many years they would have to make these adjustments every now and then.
