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.