Is writing unit tests a good way to start contributing to Bitcoin core?
Jumping straight to writing unit test can be hard, I suggest You check the official github page for how to contribute to the bitcoin core:

https://github.com/bitcoin/bitcoin/blob/master/CONTRIBUTING.md

If I wanted to contribute I would personally start by trying to solve some of the problems on their issues page because it is fresh, there is a lot of work, some can be easy and help is always needed:

https://github.com/bitcoin/bitcoin/issues