Hello!
That is an excellent question and I will do my best to answer it. Until the 1950s, when the first artificial satellites were launched into space, scientific research on the Sun, solar activity, and Earth's magnetosphere were restricted to the kinds of things that can be studied using ground-based observations. You may have seen images from spacecraft like SOHO or the Solar Dynamics Observatory showing what the Sun looks like in different wavelengths of light. Some of these wavelengths are blocked by the Earth's atmosphere, so solar observations before the space age focused primarily on the visual part of the electromagnetic spectrum. Even if you can only use visual wavelengths, you can still cound the number of sunspots or use spectrographs to examine the details of the visual part of the Sun's spectrum. Here on Earth, you can also study the Sun's corona during solar eclipses, when the Moon blocks out the light from the disk of the Sun, in much the same way the coronograph on SOHO does. Scientists could also make visual observations from the ground of the spectrum of the aurora or northern lights and monitor perturbations to the Earth's magnetic field on the ground.
I don't personally have any experience working with sunspot numbers or other observations from before the space age. However, I am aware of the difficulties in comparing with results from other scientists. These difficulties exist even today, with modern instrumentation. When comparing new results with previous ones, you need to make sure you understand the methods used to make the older observations and the errors introduced by these techniques. Fortunately, a good scientist always documents his or her work, and publishes books or papers describing the observations made, the instrumentation used, and their interpretation of the data. You can learn a bit more about how Galileo made his early solar observations here:
http://galileo.rice.edu/index.html
The graph you saw showing solar activity that stops at 2000 is probably a little bit out of date. You can see more recent plots of the sunspot number that include data from 2010 and 2011 here:
http://solarscience.msfc....v/SunspotCycle.shtml
http://www.swpc.noaa.gov/SolarCycle/
and the actual sunspot number data scientists used to make this plot here:
http://www.ngdc.noaa.gov/...&s=1&d=8,4,9
Kris