Well Cristo gave you a good starting-point link. We are talking about many different sorts of measurement, millions of datapoints, compendia of different sorts of data that must be fitted together consistently.
If you are motivated to learn about the observational bases of cosmology you can start nibbling away. Pick some topic like this (WGL):
learn about it and then move on.
Or pick a more general topic. For instance: how is the mass
of remote objects determined?
Largely by orbit speeds and statistical multibody analogs of orbit speed. So in that case what is actually measured is very often a doppler shift
. If you look at a spiral galaxy edge-on and see that the righthand edge is coming towards you at some km/s speed and the lefthand is going away at the same km/s then you have a handle on the mass of that galaxy.
Then you look at a cluster of galaxies and measure the individual dopplershift speeds of separate galaxies and estimate how much mass the cluster has to have in order to keep from flying apart.
And the mass of the cluster is much bigger than the sum of the individual masses. And you account for the difference in various ways and the components can be seen.
The dark matter cloud associated with the cluster may be seen and mapped by WGL. A kind of contour map is made showing the distribution of dark matter density. That contributes a lot of the mass but there is also a thin hot plasma called the Intergalactic Medium (IGM) which can be seen by Xray telescopes. So the IGM can also be mapped.
Of course masses of individual stars are told basically by orbit speed, with an overlay of other indicators that were originally calibrated by orbit speed masses.
So what are we measuring?
Lightcurves (fluctuations in brightness over the course of days)
Absolute brightness compared to color (spectra)
Distance (measured on various scales with various techniques)
Cosmological redshifts (factor by which distances have increased while light was in transit)
Masses (by observing dynamics and by WGL and by Xray and radioastronomy observation of dilute media and other inference tools)
I can't easily list all the different "senses" that astronomers have. They use a complex web of inference, somewhat like detectives reconstructing
what must have happened in order for us to be getting the signals that we are getting.
Of course the CMB is a big deal. The temperature map. The statistical size of the flecks and blotches of temperature variation. Another big deal is socalled galaxy redshift surveys where you simply map the distribution of galaxies in a huge volume of space and find ripples (and other wisps and cobwebs of structure). Then you have to explain dynamically how these wisps condensed from ripples etc etc. Lovely business. Currently used code letters for that kind of data and inference is "BAO" (baryon acoustic oscillation).
And supernovae are a big deal (SN) which means measuring the brightness and lightcurve (variation of brightness over several days) and the spectra (color rainbows).
It is hard to think of all the kinds of measurement at once. Hard for me, anyway. All intricately interrelated.
If you go to sources and you see a table of cosmo parameters arranged in columns depending on what data was fitted to, look for the column labeled "WMAP+BAO+SN" because that means the numbers come by fitting to all three main kinds of data---the WMAP map of the CMB, and the overall galaxy count survey, and the supernovae.
You are right about the model parameters. They are not the nittygritty that one measures. One measures a whole lot of fascinating stuff by Xray, and UV and optical and infrared and microwave and radio and simple clock timing of day by day variation. And one makes maps and measures sizes and angles on maps. And then one fits it as best one can with the best model one can think of. Only at then end come the parameters that give the best fit.
Along those same lines, I suspect if you want to learn cosmology it makes better sense to start (not with some abstract bestfit parameter but) with the gritty detail of how do astronomers determine distances, and masses. Also I didn't mention the Friedmann Equations, which derive from the best theory of gravity we have so far and are the bedrock. These two equations relate the overall degree of flatness and the rate of expansion to the average density. (And pressur in situations where that plays a role.)
In case you're curious here is a look in the engine room or at the level of the factory floor, where you see tables like Table 2 on page 4, that have bestfit parameters labeled "WMAP+BAO+SN".
Latest and greatest, indigestible, incomprehensible, not intended for public consumption. Notice that Ned Wright is one of the authors. He has a website.