I find the parallax effect parallax effect especially convincing evidence. Parallax is the shifting of lines of sight due to translation, eg by waiting half an earth year at which point theory tells we have moved about 16 light minutes around the sun from where we were.

Regarding retrograde motions: as Ilya said, Kepler's 2nd law closer planets move faster. Now draw two circles centered at Sun' with a point Earth' moving on the inner circle and another point Mars' moving more slowly but in the same sense, say counterclockwise on the outer circle. Drow a line between the two moving points. That line indicates how Mars looks, viewed from earth, relative to the distant stars. How does the line move? Put the Sun at the origin. If the order is Sun-Earth-Mars, with Earth and Mars on the positive x axis, then the slope of the line is decreasing. But put the order Earth-Sun-Mars with Earth on the negative x axis, Mars on the positive x -axis. The slope of said line is now increasing. One is prograde' the other `retrograde'.

Finally, the explanation of elliptical versus circular motion is more of an Occam's razor business. Originally we had
Ptolemy's ``epicycles''-- in essence Fourier series. Ptolemy had earth at the solar system center and each planet moving on a system of nested circles, as in $z(t) = r_1 e ^{i \omega_1 t} + r_2 e^{i \omega_2 t} + r_3 e^{i \omega_3 t} + ... $, $r_1 > r_2 > \ldots $. Ptolemy needed 20 to 30 circles to account for observations. Kepler realized that but putting the sun at the center' and having the planets move in slightly eccentric ellipses with focus, a bit off from the sun, sweeping out equal areas in equal times'', he could account for all of Ptolemy's data plus Brahe's much more detailed data.

I find the parallax effect parallax effect especially convincing evidence. Parallax is the shifting of lines of sight due to translation, eg by waiting half an earth year at which point theory tells we have moved about 16 light minutes around the sun from where we were.

Regarding retrograde motions: as Ilya said, Kepler's 2nd law closer planets move faster. Now draw two circles centered at 'Sun' with a point 'Earth' moving on the inner circle and another point 'Mars' moving more slowly but in the same sense, say counterclockwise on the outer circle. Drow a line between the two moving points. That line indicates how Mars looks, viewed from earth, relative to the distant stars.
How does the line move? Put the Sun at the origin. If the order is Sun-Earth-Mars, with Earth and Mars on the positive x axis, then the slope of the line is decreasing. But put the order Earth-Sun-Mars with Earth on the negative x axis, Mars on the positive x -axis. The slope of said line is now increasing. One is `prograde' the other 'retrograde'.

Finally, the explanation of elliptical versus circular motion is more of an Occam's razor business. Originally we had
Ptolemy's ''epicycles''-- in essence Fourier series. Ptolemy had earth at the solar system center and each planet moving on a system of nested circles, as in $z(t) = r_1 e ^{i \omega_1 t} + r_2 e^{i \omega_2 t} + r_3 e^{i \omega_3 t} + ... $, $r_1 > r_2 > \ldots $. Ptolemy needed 20 to 30 circles to account for observations. Kepler realized that but putting the sun at the 'center' and having the planets move in slightly eccentric ellipses with focus, a bit off from the sun, sweeping out ''equal areas in equal times'', he could account for all of Ptolemy's data plus Brahe's much more detailed data.