This week's book giveaway is in the OO, Patterns, UML and Refactoring forum. We're giving away four copies of Refactoring for Software Design Smells: Managing Technical Debt and have Girish Suryanarayana, Ganesh Samarthyam & Tushar Sharma on-line! See this thread for details.
IMHO, it's best to get a few smaller pieces working front to back, then move on to the next piece, rather than building one whole layer before moving on to the next layer. Then once you've got a couple pieces proven out vertically, you may want to go flesh out larger chunks of a couple of lower layers before doing more end-to-end work. I find that alternating between vertical and horizontal work helps me keep a more holistic picture of the project in mind and also helps flesh out design problems and stuff I just plain didn't think about sooner rather than later, which makes them easier to fix.
Note that I'm talking about software development in general, not Android in particular.