First I tried testing models and controllers, skipping views. Now I use integration tests with Capybara because it effectively tests the whole stack without writing redundant test code.
What's not clear is the pros and cons of each approach. Over time you end up using your own approach that works best for you. It's one thing to know that you need to "test your code", but with all the various testing libraries created, it can quickly become overwhelming.
In an ideal world, I'd like to learn more about the various approaches. Test::Unit vs Rspec, vs all the additional testing add-ons and how everything works together. When you end up spending just as much time writing tests as you do writing code, you quickly realize how critical getting the right approach to testing really is.