Already on earth the time is a bit of a problem with the time-zones. It is not that big a problem though, since we all "travel" at (nearly) the same speed.
Not so in space. With the time dilatation (and probably some other factors) one cannot simply rely on ones watch. So, how would it be done?
With the time dilatation (and probably some other factors) one cannot simply rely on ones watch.
Ultimately, a "perfect watch" is all one can rely upon. A good approximation of a "perfect watch" is an atomic clock that is located at the center of mass of your (relativistic) spacecraft.
The basic problem is that simultaneity is relative even in special relativity. In general relativity, the simultaneity of relativity is a manyfold aggravated issue. Ben Franklin's statement that "Three can keep a secret, if two of them are dead" is very applicable to relativistic mechanics. How do you keep time, synchronously? As long as more than one are trying to do it, you can't.
We can (kind of) pretend that relativistic effects are linear, and in that case you end up with something like JPL's ephemeris time Teph or the IERS's/IAU's TCB (barycentric coordinate time) or TDB (barycentric dynamic time). None of them are perfect; they can't be.
For the kind of space travel we are performing in this day and age, time dilation effects are too small to be of much concern for mission planning. The time coordination of most space missions is synchronized to the time at the mission control. The International Space Station does its scheduling and daily routine according to Coordinated Universal Time (UTC).
When you would perform space travel with relativistic speeds (note that we are entering the realm of science fiction here), you have to consider what goal you want to reach with your time-keeping. When you need to check on your fusion reactor temperature every hour, you would do so according to a conventional clock in your own reference frame, because your reactor wouldn't care about how much time has passed on Earth. But when mission control back on Earth expects hourly status reports from you, you would have to separately track the time according to Earth's frame of reference to ensure that they arrive on time. Your earth-clock would have to be aware of your current speed in earth's frame of reference so it can adjust its speed accordingly. Also keep in mind that information can only travel with the speed of light, so the earth-clock would also need to be aware of your current distance from Earth (in Earths reference frame), and add that number of light-seconds to the time it displays.
What exactly is being measured?
If it's one's own timeline while travelling, then a wristwatch is fine.
If it's mis-synchronization of clocks that are distant and possibly moving, then one clock must be chosen as a standard, either distant and unchangeable (pulsars) or local and convenient (an accurate clock in Greenwich).
