Earth doesn't need to fear the black hole at the center of our galaxy. A more likely threat to our solar system is the Andromeda Galaxy, 2 million light years away and racing towards us at nearly 700,000 miles per hour. The collision may begin in a few billion years, with a couple more (billion) to finish. But even then, physicists think our solar system is likely to survive the collision - or get flung out into more lonely space.
From our standpoint, though, long before the merger starts, our dying sun will begin expanding, making Earth too hot for life. Hopefully, we will have long since departed - watching the merger from some safe - and distant - location. :)
Nova did a special on it a while back: http://www.pbs.org/wgbh/nova/space/monster-milky-way.html