For the purpose of this discussion, let's assume that faster-than-light (FTL) travel is possible. Let us also assume that we are just beginning to explore the galaxy with FTL-capable spaceships.
How would we navigate the galaxy?
Today, we have GPS to help us navigate around the world. But in our scenario, there is no galactic GPS system until we actually go out there and build one. That means in the early stages of our galactic civilization, we need to navigate the galaxy without a galactic-positioning-system in order to build up our GPS network.
In the olden days before GPS, we navigated the seas using the stars. By locating the North Star, and measuring its position in the night sky, we can calculate our approximate latitude. To measure longitude, we need to be able to keep accurate track of time, and we need to know our speed. Using these techniques, early sailors could chart the ocean and create a map of the world.
The techniques of old cannot be used for galactic navigation. When we look at the stars, we are looking into the past. The distances between stars are so great that light from the opposite side of the Milky Way Galaxy would take a hundred thousand years to reach us. In that time, the stars would have travelled to a completely different location within the galaxy, and some stars might have already died out. Thus, the constellation in one part of the galaxy will look radically different from the constellation in another part of the galaxy. If we use our FTL-drives to jump from one part of the galaxy to another, the constellation that we know would be inaccurate. The further away from Earth that we jump, the more inaccurate our constellation would be. Jump far enough away, and it would be impossible to figure out which star is our Sun. Even if we could find it, we would be looking into the past, so its present location in the galaxy would be unknown to us. We would be lost.
Let's also not forget that time is necessarily distorted when we travel faster than light. Close to the speed of light, we will experience significant time dilation. Time will be much slower for us than it is for people on Earth. This means we cannot use a simple calculation of speed vs time to calculate how far we've travelled. Of course, we cannot travel faster than light in normal space-time, so we'd probably have to create some kind of space-time bubble around our ship so that these time-dilation effects might not come into play. Still, how would we measure how quickly our space-time bubble is travelling through the galaxy? Is the time in our bubble the same as time outside the bubble? If we don't know the time, we cannot calculate where in the galaxy we will end up.
I think the most sensible solution is to chart out the galaxy one star at a time. We travel to the closest stars to our own. If we only travel within a few light years of Earth, our Earth-based constellation would still be accurate enough for us to infer where the Sun should be. We then set up a space-station and measure the star's actual position relative to Earth. These stations beam this information back to Earth, which will act as the central hub that will update all the space-stations around all the stars we've explored. The information transmission between the various space-stations will occur instantaneously using quantum-entanglement. We gradually explore stars that are further and further away from Earth, and methodically build up our galactic-positioning-system.
Once we have our GPS network set up, we will know the actual position of all the stars within the network, and this information will be updated in 'real time'. Now we can travel to any star within the network and rest assured that we will arrive at the right place and the right time. And after we're finished with our trip, we will be able to find our way back to Earth. Of course, setting up this network will be extremely expensive. If other alien civilizations discover our network, and wish to make use of it, we should charge them a lot of money.