I spent quite a lot of time many years ago doing my own independent checks on astronomy.
I started down this line after an argument with a friend who believed in astrology. It became apparent that they were talking about planets being in different constellations to the ones I'd seen them in. I forget the details of their particular brand of astrology, but they had an algorithm for calculating a sort-of 'logical' position of the planets in the 12 zodiacal signs, and this algorithm did not match observation, even given that the zodiacal signs do not line up neatly with modern constellations. They were scornful that I was unable to tell them where, say, Venus would be in 12 years time, or where it was when I was born.
So challenged, I set to.
The scientific algorithms for doing this are not entirely trivial. I got hold of a copy of Jean Meeus' Astronomical Algorithms, and it took me quite a lot of work to understand them, and then even longer to implement them so I could answer that sort of question. They are hopelessly and messily empirical (which I take as a good sign) - there is a daunting number of coefficients. Eventually I got it working, and could match observation to prediction of planetary positions to my satisfaction - when I looked at them, the planets were where my calculations said they should be, more or less.
It's hard with amateur equipment to measure accurate locations in the sky (e.g. how high and in which direction is a particular star at a particular time), but relative ones are much easier (e.g. how close is Venus to a particular star at a particular time). The gold standard for this sort of stuff is occultations - where you predict that a planet will occult (pass in front of) a star. There weren't any of those happening around the time I was doing it, but I was able to verify the calculations for other occultations that people had observed (and photographed) at the date and times I had calculated.
These days, software to calculate this stuff - and to visualise it, which I never managed - is widely available. There are many smartphone apps that will show you these calculations overlaid on to the sky when you hold your phone up to it. (Although IME their absolute accuracy isn't brilliant, which I think is due to the orientation sensors being not that good.) This makes checking these sorts of predictions very, very easy. Although of course you can't check that there isn't, say, a team of astronomers making observations and regularly adjusting the data that gets to your phone.
I was also able to independently replicate enough of Fred Espenak's NASA eclipse calculations to completely convince me he was right. (After I found several bugs in my own code.) Perhaps the most spectacular verification was replicating the calculations for the solar eclipse of 11 August 1999. I was also able to travel to the path of totality in France, and it turned up slap on time and in place. This was amazing, and I strongly urge anyone reading this to make the effort to travel to the path of totality of any eclipse they can.
Until I'd played around with these calculations, I hadn't appreciated just how spectacularly accurate they have to be. You only need a teeny-tiny error in the locations of the Sun/Moon/Earth system for the shadow cast by the moon on the Earth to be in a very different place.
I also replicated the calculations for the transit of Venus in 2004. I was able to observe it, and it took place exactly as predicted so far as I was able to measure - to within, say, 10 seconds or so. (I didn't replicate the calculations for the transit in 2012 - no time and I'd forgotten about how my ghastly mess of code worked - and I wasn't able to observe it either, since it was cloudy where I was at the time.)
More recently, you can calculate Iridium flares and ISS transits. Again, you have to be extremely accurate in calculations to be able to predict where they will occur, and they turn up as promised (except when it's cloudy). And again, there are plenty of websites and apps that will do the calculations for you. With a pair of high-magnification binoculars you can even see that the ISS isn't round.
All this isn't complete and perfect verification. But it's pretty good Bayesian evidence in the direction that all that stuff about orbits and satellites is true.
Wow.
The closest analogue I have to that is grabbing planet positions and velocities from JPL's HORIZONS system, then doing small time steps holding accelerations constant.
That's how I know the (mathematical) solar system behaves as claimed. Except that Mercury's orbit will eventually become so elliptical and gain so much energy that it careens in and out of the solar system until it flies off to infinity (or people are also right about the limitations of the approximation technique I was using).
Suppose you distrusted everything you had ever read about science. How much of modern scientific knowledge could you verify for yourself, using only your own senses and the sort of equipment you could easily obtain? How about if you accept third-party evidence when many thousands of people can easily check the facts?
My purpose with the question isn't to cast radical doubt on science; rather, it's an entertaining game of trying to understand how we know what we know. Thinking through these sorts of questions also helped me notice interesting things in the history of science that I hadn't previously focused on. It might also be of interest from a science education perspective.
Some things are much easier to check than they used to be. As late as the 19th century, there were people who were publicly skeptical about the curvature of the earth. Skeptics and scientists did careful measurements (notably the Bedford Level Experiment) to observe the earth's curvature. Today, you can verify it by phoning a friend a few time zones away and noticing that the sun reaches the zenith at steadily later times as you move west. This only makes sense if the earth is curved.
Some things are still hard to check. I don't know an easy way to show that the Earth orbits the Sun. The direct way to show it would be to measure stellar parallax. But even the closest stars have a parallax of less than an arcsecond. My understanding is that very few amateurs are able to take measurements with that level of precision.
Some things are surprisingly easy. There are lots of easily accessible demonstrations of quantum phenomena. For example, a ten dollar spectroscope will show you that an incandescent light bulb has a continuous spectrum, and that LEDs and fluorescent bulbs don't. Bright-line spectra are very much a quantum mechanical phenomenon -- it's a sign that the atoms in the light source have fixed energy transition levels. Spectroscopy was one of the key early lines of evidence for quantum mechanics, and it blows my mind that it's something you can just see whenever you want, with a negligible equipment cost.
Pretty much all of modern chemistry and solid state physics rests on a quantum foundation, and you can test a great deal of chemistry pretty easily. If you are in doubt that water is a bonded compound of two gasses, you can do the electrolysis very easily yourself. You can observe the periodicity of chemical elements yourself if you buy alkali metals (don't try this one at home!). If you are willing to accept slightly indirect evidence, the entire semiconductor industry is about precisely controlling the conductivity of impure silicon, and this would make no sense if quantum mechanics weren't a reliable guide to electron energy levels in the solid state.
I don't feel quite as qualified to play this game for biology. I imagine that antibiotic resistance is a well-enough documented case of evolution through natural selection to serve at least as a proof of concept. DNA sequence comparisons across species are emphatic evidence of taxonomic trees, if you trust the scientists not to be part of a vast conspiracy.
It feels almost impossible that it's easier to see quantum mechanical effects than it is to see that the earth orbits the sun, but it does seem that way.
Some questions: