How to write Unit test during development?

Unit test is good for maintaining a software, especially if the maintainer is not very familiar with the whole system.

But I have a question here:

Before I complete the whole software system, how should I determine which functions I need write unit test for? In other word, what's the best granularity for unit test?

What's worse, The name or functionality of tested functions may be changed during the develop process or after time-to-time refactors, how should I maintain the unit test?