I am wanting to learn everything there is to learn about the Sun from our solar system to the ends of the universe.
Ok... some of Vojta's info is pointing in the right direction.
Via spectroscopy we can get a very good estimate on the chemical composition of the sun.
The mass of the sun can be deduced from Newton's laws (F = G Mm / r^2) and the period of the earth's orbit. Technically, you also need the mass of the earth, but it is so insignificant as compared to the sun that it's negligible.
We can measure the amount of energy received by the earth (which is the flux received by the earth). Since we know the earth diameter and the distance to the sun, we can compute how much energy the sun is sending out (and hence generating -- assuming its in equilibrium) each second (3.8 x 10^33 ergs/second or 3.8 × 10^26 watts).
Those are some of the basics, but I'm sure if you google "Sun" you'll uncover a wealth of information.
Well we can use for example spectroscopy, this means that we can know what is the surface made up of.
There is radiation coming from Sun, solar wind it is called. We can measure its speed and learn more about how it was created.
Of course we can just observe it with normal telescope. After astronomers found out the value of astronomical unit (AU) which happened I think around 18th century they were also able to find out the mass of Sun.
We are able to measure incoming radiation. This was very important during Apollo missions, even now lot of people think that there is no way to get through Van Allen radiation belts because of particles coming to us from Sun.
We learned that there are sun-spots in periods of 11 years.
Also during the measurement of radiation we were able to find out how strong events can solar flares be. Actually if one of the very strong ones hit us, we would have extreme black out (there is page which is predicting space weather but I do not remember its name).
We also found out using physics and particle accelerators that elements can fuse together. We learned that there is evolution in stars life and by observation in space we found out that there is going to be white dwarf instead of Sun in several billions of years.
We found that there is solar corona around Sun which is very hot but nobody is sure where it gets its energy.
Also we can see in our Milkyway how stars are created so we know that there must have been disc of matter which happened to create Sun.
There is much more to this but this is what I was able to think of right now.