I'm a big fan of "economic testing". Test that are not going to "be economically viable" to write during the dev't of new software, usually do not get my permission to be written in that phase.
Unit test thus get written for part that we absolutely want to be maximum certain of that it works correctly. Mostly that functionality can be extracted into a library. For instance a lib that deal with scheduling of events of arbitrary length: we do not want to have issues with those getting messed up, so we try to get 100% unit test coverage of the functionality in that lib.
Another thing with unit test is that I found they mean something different in dynamic languages than in strongly typed languages. In the first unit test are often something to provide confidence that the type system can give you in the latter. Sure both will also test the logic. But this understanding has pushed me to love strong types (as a rubyist), as I can go with less unit test and still be more confident about my code base when it is being heavily worked on. Refactoring becomes so much easier with strong types, as all is "in the code".
When it comes to integration tests I usually want to make a biz case for them. How much does it cost to have it all manually tested, and how much do the automatic integration tests cost to make. While including the benefit of being able to run the automated integration tests at near zero cost (thus several times per day if we need to).