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Navigation requires a reference frame. We need reference frames to tell us where we are with respect to other objects and we need reference frames to tell us how we are oriented with respect to other objects. There is no single universal frame that is used for all operations. Typically, the reference frame that most simplifies the mathematics and visualization is used for a particular objective. For example, just for the International Space Station (ISS), we use more than 15 reference frames.
There is a reference frame that is helpful for pinpointing a location within the ISS. It is Cartesian and gives locations with respect to forward and aft, starboard and port, overhead and deck. Another frame is used for structural calculations and gives ISS hardware locations in terms of X, Y, and Z. But since that body frame is with respect to the center of mass of the ISS, it doesn't really help us understand the orientation of the vehicle, so we need an additional frame for that, one that can take that X, Y, Z body frame and describe it as rotations about the local vertical and local horizontal. And, although that frame (LVLH) is great for telling us the vehicle attitude, it isn't helpful for telling us the vehicle location. We need a frame for that, too. Most commonly we use a frame called J2000 to describe the location of the vehicle. J2000 has its origin at the center of the Earth. As the ISS moves forward in its orbit and as its altitude changes because of either decay or thruster burns, J2000 can provide a precise understanding of the ISS location with respect to the Earth, below. It gets a bit more complicated, though. Individual systems and sensors often use their own frames and then the software converts from one to another. For example, GPS provides the ISS location in the CTRS (Conventional Terrestrial Reference System) and the software converts that to J2000 for use by the vehicle GNC software and by the flight controllers.
But, now we're getting to the heart of your question. What happens when a spacecraft is no longer in Earth orbit? How do we define its location, then? The answer is to identify the primary body that the vehicle is respondent to. For spacecraft traveling throughout the solar system, the ideal reference frame is one centered on the Sun. But, once the spacecraft is close to its destination (e.g. Saturn), it would make sense to also begin to report coordinates with respect to that body.
And what about when we leave the solar system? Well, we haven't really done that yet. Although Voyager has technically left the solar system, it still makes sense to describe its location with respect to the Sun.
As we move far beyond the Sun, we would need to look for a frame that optimizes the situation. It might make sense to use a frame that is centered on the center of our galaxy, because our Sun, the spacecraft, and whatever star we are going to visit will all be in orbit about the center of the galaxy.
But, currently we care about how things are located with respect to our solar system. For that, a frame called ICRS (International Celestial Reference System) is commonly used. It is a quasi-inertial frame centered on the barycenter of our solar system. It is maintained by tracking the positions of almost three hundred sources outside our galaxy (e.g. quasars).
While coordinates of a frame like J2000 are given in cartesian X, Y, Z, a frame like ICRS uses the spherical coordinate system of Right Ascension, Declination, and Distance.