I find the Shimazu a great clan to play if I want to play a combat-focused game--rapid expansion. I don't play them like I would play the Uesugi or the Hojo. Because of their location, if you attack fast, furious, and constantly, it's pretty easy to sweep across the board. I don't spend much on economic improvements, and only a bit on military improvements. I increase income by increasing territory, fast. I do build a lot of ports. This strategy also helps to get at the Uesugi and/or Hojo before they can achieve "horde" proportions. If I were to play a "how fast can I take over Japan" campaign, I'd probably try the Shimazu first, because they are geared for that kind of game (at least, they seem so to me).

The Shimazu's special unit (No Dachi), and the requirements to produce it, also encourage this approach.