If you allow indirect evidence, then the rotation of the earth around the sun follows from the seasons which are easy to observe. Or rather from the earthes tilt. Given that you confirmed a) constant (almost) distance to the sun, b) conservation of angular momentum (via observation of spinning tops), c) changing angle of midday sunlight thru the year the only remaining conclusion is that the earth circles the sun.
Almost. you have to exclude that precession of the eath is causing the changing tilt. You'd have to estimate the amount of precession (http://en.wikipedia.org/wiki/Precession#Classical_.28Newtonian.29) which should be possible given some rough estimates of the mass of the earth.
Heliocentrism isn't the only theory that implies the existence of the seasons:
...The motion of the sun eastward along the ecliptic combined with the rotation of the stars westward along the equator to account for seasonal changes in the length of daylight, while moving the circle of the sun's motion slightly off-center toward Gemini adjusted for the inequality of the seasons. The resulting extremes of apogee and perigee of the sun also explained for writers on the sphere why the arctic regions are too cold, and the southern hemisphere is too hot, to be habi
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: