and study the solar system and universe relative to it and why not relative to the Earth?
When you're trying to understand the mechanics of a system it's usually convenient to choose coordinates that reflect the symmetry of the system. The solar system is roughly centrally symmetric because the Sun is by far the largest mass in it, and the coordinates that reflect this symmetry are polar coordinates with the Sun at the centre.
For example in these coordinates if the Earth was the only object apart from the Sun, the Earth's orbit would be (nearly) a ellipse. The presence of the other planets (mainly Jupiter) perturbs the Earth's orbit, but we can handle this by perturbation theory starting with the elliptical orbit and adding on the perturbations caused by the other planets.
So taking the Sun as a reference point is a reflection of the symmetry of the Solar system.
As noted in other answers, if we're describing the galaxy the Sun is no longer the best place to set the origin of our coordinate system, and we'd use polar coordinates centred on the centre of symmetry of the galaxy. Likewise to describe a galaxy cluster we'd choose the origin to be the centre of mass of the cluster. At the very largest scales the universe is isotropic and homogenous, so it doesn't matter where we place the origin.
It's all about the context in which you want to analyze particular issue.
If you are studying the solar system, the most suitable, would be to consider the sun as the center of the system.
If you are studying the Milky Way, the sun is not a good reference point, you should take the center of the galaxy.
Similarly, to locate the stars from an observer on earth, the "celestial sphere" is used, and that does not mean returning to the model of Ptolemy.
It's all about the scope that is intended in a particular analysis.
why not relative to the Earth?
Scientists do express things relative to the Earth, where that makes sense. I couldn't imagine trying to forecast the weather or model the global circulation of the Earth's atmosphere from the perspective of a non-rotating frame with it's origin at the solar system barycenter. Astronomers, at least those dealing with Earth-bound equipment also have to express things relative to the rotating Earth. They need to know where to point their Earth-bound equipment, after all.
Meteorologists not only represent the weather in terms of a geocentric point of view, they model the weather from that perspective. This means that all kinds of fictitious forces arise in their models because the Earth is both accelerating and rotating. There's nothing wrong with that per se, so long as one does the mathematics correctly. Since any alternative (e.g., a non-rotating frame) is even worse, meteorologists make sure they do the mathematics correctly.
On the other hand, using an Earth-centered, Earth-fixed perspective to describe the behaviors of the distant objects that astronomers observe is even more ludicrous than is trying to describe the Earth's weather from the perspective of a non-rotating, barycenter-based perspective. It's much easier to describe the behaviors of solar system bodies from the perspective of a non-rotating, barycenter-based frame. That frame similarly isn't all that good for describing the behavior of the galaxy as a whole.
A reference frame at rest with the Sun is, with a good approximation, an inertial system (much better than one at rest with our planet or other bodies in the Solar system, essentially in view of the hugely larger mass of the Sun). Physics in inertial reference frames has the simplest form. For instance the motion of planets around the Sun is described along ellipses with the Sun as one of the center, with a good approximation. The ultimate reason of this fact (assuming the Newtonian form of the gravitational law) is that I pointed out: If referring to another reference frame one has to include so-called, in a sense unphysical, inertial forces in addition to the gravitational one to explain the complicated motion of planets. All this reasoning makes sense if disregarding cosmological issues where general relativity plays a crucial role, and instead adopting the Newtonian paradigm.
People who are studying something something generally use a frame of reference which is reasonably close to the things of interest. While it might in theory be possible to measure the stature of a man by very accurately determining the distance from the center of the Earth to the bottoms of his feet, as well as the distance from the center of the earth to the top of his head, and subtracting the first from the second, it is both easier and more accurate to measure the height of the top of the person's head relative to that of a surface upon which the person is standing.
The distance from the Earth to the Sun is so much greater than the diameter of the earth that when reporting the locations of terrestrial objects, it makes sense to describe locations relative to other terrestrial objects. Only if one is regarding everything on Earth as a single point does it make sense to use the Sun as a reference point. Going the other way, the distance between the Sun and the nearest star is so much greater than the distance between the Sun and the most distant known orbiting body that it only makes sense to use any reference point outside the solar system if the Sun and everything orbiting it are regarded as a single point.
Because of GPS and similar technologies to determine the locations of terrestrial objects, determining a person's stature by measuring the absolute positions of the head and feet would almost be practical; from an astronomical perspective, however, trying to use any sort of solar or galactic coordinate system to plot the locations of terrestrial objects would be like trying to use a barometer to measure stature. A barometer may allow rough determinations of altitude, but the uncertainty in the altitude measurements of the person's head and feet would likely exceed the distance between them by orders of magnitude, making any stature determinations meaningless.