If you brew gongfu, yixing is definately the way to go, but it isn't really any better than porcelain for western brewing.
Since you said you brew green tea, you should consider getting a gaiwan. Yixing are good for teas that brew at high temperatures, but not for green tea which likes cooler water.
You can brew the tea in the gaiwan, and drink directly from the gaiwan with the leaves in it, or you can pour it out. I usually brew and drink from the gaiwan because it's a really simple method, and makes a good cup of tea.
And no, it usually doesn't get bitter sitting in the gaiwan the entire time. If it becomes bitter, you either used too much leaves, or too hot of water.