Yes, the Sun, and all the planets of the Solar System, are all part of the Milky Way galaxy. The Sun is but one of 200 billion stars in the Milky Way. Most of these are not visible in the night sky. In fact, only about 9,000 nearby stars are visible to the unaided eye.

We exist in the outer parts of the galaxy, in the "galactic suburbs," as it were. We're about 26,000 light-years from the center of the Galaxy. We revolve around the Galaxy about once every 225 billion years. Since the Sun formed 5 billion years ago, we have made about 22 revolutions around the Galaxy. Since the dawn of human origins, the Sun has only revolved 1/1250th of its orbit around the Galaxy.

First, rest assured that the Sun won't explode. Low mass stars like the Sun toss off their outer layers with little violence, leaving behind a white dwarf.

The quickest that news of anything happening on the Sun comes to us is at the speed of light, which takes about 8 minutes 20 seconds to reach the Earth. If we lived around a more massive star that was going to explode, we may get a little advanced warning (perhaps a couple of hours) from tiny particles called neutrinos before the flash and blast.