Isn't astronomy interesting?
The sun is more than just a familiar security blanket. In a way, it's the glue that holds our solar system together. The thought of losing our sun is terrifying.
So, what would happen if our sun was gone?
Imagine the Earth was a block and a space craft is traveling parallel to one surface at a high rate of speed. Meanwhile, gravity is pulling it down towards the block, so its path is really at an angle, heading towards the block. However, by the time it reaches the plane of that side of the block, it is beyond the edge, and heading away from the block.
Once again, gravity is pulling it toward the block, but its speed keeps that fall from being a straight line to the face of the block. In other words, it is falling AROUND the block.
The idea of an orbit is to make sure your forward motion matches the downward motion caused by gravity.
I hope that makes sense. So, the orbiting object is trying to move in a straight line, but gravity makes its path circular. If there was nothing there (like the sun), an object would continue to move in a straight line. If our sun disappeared, the planets would all fly off into space.
Now, other things could happen, depending on their relative positions, but those scenarios are too numerous and too complicated to discuss. Their paths outward into the galaxy would be affected by each other's gravity, but I doubt enough to change their directions radically. Eventually (say millions of years), each would probably take up orbit around other stars, unless they collided with another object.
Did you know that astronauts in orbit are really not weightless? They are affected by gravity, too. However, they are falling at the same rate of speed as their spacecraft, so there is no force to hold them against any one surface. It gives the feeling of weightlessness, but it isn't.