How did people try to study space in the past?

The most ancient is by observation, structures were built that aligned to mark the movements of the celestial bodies from solstices to those that could gather data to predict a solar eclipse. Examples of such are the structures built by the Babylonians, (go to Wikepedia-Babylonian astronomy) who mathematical methods to take & process the data empirically gathered to build an impressive understanding of planetary movements. Stone henge is also thought to be an observatory & physical map too. The Greeks were able by geometry to calculate the diameter of the earth, but explored & tested many models of the solar system from a central earth to a one as we have now. They also developed astronomical calculation devices of which one is reputed to be the first analog computer Antikythera mechanism.
The next surge in exploration came with the Renaissance with Nicolaus Copernicus who first developed a mathematically predictive heliocentric system (we & the planets go around the sun!!!) Galileo Galile built & enhanced upon this model with his invention, the telescope.
Newton & Pascal went forward to derive new maths and establish the physical laws that would enable the heliocentric system to work so further our understanding.
Before we finally go into space Albert Einstein through observations derived & proved relativity, the properties of time/space including the absolute speed of light. Which allowed In 1919, American astronomer Edwin Hubble, to establish that those faint patches of light where not nebula, but Galaxies and by applying the absolute speed of light estimate there distance away but applying the Doppler effect to the spectrum of their light establish they were moving ( the further away they were the faster they were going away (more low frequency red (red shift) interestingly, establishing an expanding universe & the logical possibility of it starting from one point).
Add the invention of radio & their telescopes, nuclear physics as from Rutherford & here we are!

My advice is to Google most of the above as I have summarized a lot, missed out the Mayans & Egyptian contributions amongst many so do not claim this to be a definitive record of major milestones.
Hope these notes help & not too late.