What is the escape velocity of our Galaxy relative to?
One way to think of the Escape velocity is to imagine it backwards.

Instead of a rocket being fired into space, think of the same rocket, starting at rest relative to the Earth at a great distance (for this thought experiment pretend the sun, doesn't exist and the rest of the universe is empty), then let it fall to the ground. No matter how far away you start, the speed when the rocket hits the Earth will always be 11km/s or less. 

So if you start with 11km/s or more at the Earth's surface, you have enough energy to coast forever into space. The escape velocity at the surface is 11km/s, the escape velocity would be different, and lower, if you were already in a high orbit.

We can do the same thought experiment with a galaxy. We know, roughly the distribution of mass in the galaxy (most is in a large blob of dark matter  with the visible disc inside it) Now take a rocket ship a very great distance from the galaxy, and allow it to fall.  As it passes the sun, the rocket will be moving at 537km/s relative to the centre of the galaxy. (there is a black hole there, but it is tiny compared to the galaxy, so has very little effect on the gravity of the galaxy)

If you know how the mass of a galaxy is distributed we can get a good idea of how fast something would be travelling at any point, if allowed to fall. And by the same argument as above, that gives the escape velocity at that point.

The escape velocity around the sun is about 1/600 of the speed of light. The journey to leave the Milky way is at least 30000 light years long, This means it is going to take about 20 million years to reach the edge of the galaxy, since you will be getting slower as the galaxy's gravity pulls you back. Space (as has been observed) is big.