What do I have to learn to make digital wristwatch?
Microcontrollers is a very general word here.
You will need to learn the following in microcontrollers:-


Interrupts (since you will be using timer to keep track of time)
RTC (Incase you don't want to use timer and want to use more reliable source for keeping track of time)
LCD or LED interface (To display the data on screen) 


On electronics side,
You just need to know how to connect these components properly and calculating resistors in series with LED's.

Power source (long life of the battery) is one very important thing you have to take care (as pointed out by @jonk in the comment). So while designing the circuit make sure that no component is consuming more than the minimum current required.

For eg, if you plan to use LED as a display then the size of LED and the desired intensity will decide the runtime of the battery. Making the LED's glow more than detectable intensity will cause the battery drain faster, so you need to find the optimized values that work for you.

You can also learn about power down feature in controllers which save a lot of power when not in use, it will display the time only when you want it to, so LED's will be on only when required. RTC would be of great help to keep track of time when microcontroller is in power down mode.

For a basic digital watch, these should work fine.