What's the best approach to develop a cross-browser JavaScript app?

Question

As one of my summer projects I'd like to build an AJAX program on my own. Not too big. I'm new to the whole matter, however I have general knowledge in programming and a good understanding of the Web workflow.

As for W3C standards, is it wrong sticking to them as if they were a platform?

Is it a good idea to code having a single platform in mind (i.e. Firefox 3 + Win7) and make it cross-browser after the initial code is finished?

In general, to what extent is browser compatibility a problem these days?