When you move to a new planet the date & time that you'll be in is entirely different. It's not like moving to a different time zone on Earth, because the times in each country are related and moves at the same rate.
On planets that behave differently from one another (their rotation and revolution around their sun) their dates will not be related at all. Say, on Imest, it may be New Year's while it's the middle of the year on Anrakid, and the following year in Imest it may be New Year's again, but a completely different day on Anrakid. It seems confusing at first, but the inhabitants of the Piarosar system should be used to this already.
Unfortunately, the complicated nature of comparing the dates and times of different planets will also mean that organising a date to meet someone from a different planet would be challenging. It would only if you had access to some kind of technology (a kind of calculator, maybe) to figure out what date on your companion's planet would match the date of meeting on your own planet, or vice versa.