Modern astronomy begins with the telescope. Before that, no one had any way of knowing what planets looked like. Venus could have been a circular disk, a square, or a brilliantly shining hamburger - no one could tell.
What Galileo did was get hold of a Dutch invention, the telescope, and look up with it. He saw things never before seen. This is how discoveries get made and old models get overthrown.
A scientific model is a proposed explanation for some phenomenon. It is first known as a hypothesis. Scientists test their models by using them to make testable predictions about the phenomenon, then checking to see if the predictions are correct. If they are, that's good - the model passed the test. If the predictions are not correct, that's bad - the model is likely not correct. Notice that a successful prediction does NOT prove the model correct, but a series of correct predictions gives one some confidence in it.
As far as proof goes, there's only one area where you REALLY prove something, and that field is mathematics.
As far as we know, Galileo didn't invent the telescope himself. He somehow (means unknown) got his hands on a Dutch invention and later improved it. It's what Galileo did with the telescope that was significant - aim it upward into the sky and observe the objects found there. Galileo's telescope was crude by modern standards. Magnifying about 20 times, it was not adequate to see the rings of Saturn. It did provide views of the Moon and Jupiter that noone had seen before.
Galileo observed a number of important things. He noticed four small points of light apparently going around Jupiter (he was right). Those four large moons of Jupiter are called the Galilean satellites. You can see these satellites with binoculars (which Galileo didn't have). These little objects were very clearly going around Jupiter and not getting left behind as Jupiter moved. This showed that there was at least one other center of revolution in the Solar System.
He also observed that Venus obviously displayed phases just like the Moon. This was ample evidence that the Ptolemaic (geocentric) model was wrong. Here's how that works. Look at figure 1.8. We are looking at a real test of the Ptolemaic model. Using the Ptolemaic model (part b of the figure), predict what Venus should look like if one had a telescope to see it with. Venus should progress from a thin crescent to a fatter crescent, then back to a thin crescent again. Remember that the Ptolemaic model has Venus orbiting a "nothing" that always stays directly between Earth and the Sun. "Here's an animation of Venus going through its phases. Here's a really good animation from Astronomy Picture of the Day.
Galileo looked through his telescope and saw a nearly full Venus. The Ptolemaic model CANNOT account for this; it is not possible in that model. This is one real proof that the Ptolemaic model is wrong - it cannot account for the full phase of Venus.
Copernicus' model (first half of 16th century) was significant in that it proposed (correctly) that the planets orbited the Sun, not Earth. It also proposed (incorrectly) that those orbits were circular. He wasn't the first one to do this; Aristarchus (ancient Greek) had done it in the third century B.C. The only problem is that a model with circular orbits does not predict planet positions accurately. It wasn't much better than the adjusted Prolemaic model. The Copernican model is more accurate and neatly solves the problems of the full Venus and the retrograde motion of the outer planets, so it has a lot going for it. Copernicus' model is correct in concept, but needs some tweaking.
It in interesting to note how Kepler tested the Ptolemiac model by using Brahe's observations. You can find that story in this from Virginia that has a nice presentation of the history.
Look at Figure 1.11. The ellipse has the property that the sum of the distances from the point on the ellipse to the focus on the left and to the focus on the right is a constant for that ellipse. If you want to draw ellipses this way, use a loop of string around the pins - the method shown is awkward. You can look at the details about ellipses to see how they work.
Kepler described the motion of the planets in Kepler's Laws of Planetary Motion.
They are simple.
1. All orbits are ellipses. (Elegant but not obvious at the time.)
2. A planet sweeps out equal areas in equal times.
3. p² = a³ * C, where p is the orbital period of the planet, a is its mean distance from the Sun, and C is a constant which depends on the units you use. If you use Earth years for the period and Astronomical Units (AU) for the mean distance, the constant C becomes 1 and can be discarded. The law now reads
p² = a³
This is extremely easy to use. Note that the Astronomical Unit is defined as the mean Earth-Sun distance (about 93,000,000 miles). Kepler didn't know how far away the Sun was, so making the distance 1 makes things easy.
The AU was accurately and precisely measured in the last half of the 20th century. It was done by measuring the distance to Venus with radar. The AU has been measured to at least 7 significant figures. We really do know how long an AU is! A very recent value is 149,597,870 km, or about 92,750,679 miles.
If you would like some confirmation of our knowledge of the AU and how things orbit the Sun, just consider Spirit and Opportunity - the two Mars rovers. Both landed very close to the centers of their target ellipses. Such precise targeting would not be possible if we did not know how solar orbits work.