We're headed off on a world wide trip in February of '09 and were just wondering what some of the best modes of transportation throughout Asia are when it comes to getting from country to country.

Yes, this is sort of a big, broad question, but any advice would be greatly appriciated!

In Thailand, buses and budget airlines work well.

Always depends on what border crossing you are talking about. Many times it will be by bus to the border, cross on foot, by bus to the next town. Sometimes one or both trips to/from the border is a taxi (per example Kyrgyzstan - China).
In Central Asia it's possible to cross certain borders by train. You can cross from Thailand to Malaysia by train.
Malaysia to Singapore by bus.
Singapore/Malaysia to Indonesia is possible by boat.
There are boats from Indonesia to the Southern Philippines.
There are boats from China and South Korea to Japan.

And there's always flying.
If you want to stay longer than 10 days, you'll have to fly into Myanmar.