To build In-The-Sky.org, it was necessary to make precise predictions of the future movements of the planets across the sky. The planetary ephemerides produced by NASA's Jet Propulsion Laboratory (JPL) in Pasadena, California are the best source of such information that is freely available.
NASA use these ephemerides to guide their space missions to their intended destinations, and so they are remarkably accurate. The positions of the planets closest to the Earth are typically listed with errors of no more than a few kilometers – as is needed to guide spacecraft into very specific orbits around these distant bodies.
In-The-Sky.org uses DE430, which was published in 2013 and covers the years 1550 – 2650 AD. It yields the positions of the following bodies relative to the center of mass of the solar system: the Sun, Mercury, Venus, the centre of mass of the Earth–Moon system, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto. It also gives the position of the Moon relative to the centre of the Earth.
The DE430 files are freely available for download here, though the documentation of the file format is extremely limited. In summary, the ephemeris is divided into 30-day chunks, and within each 30-day chunk, a formula is given to compute the position of each planet. These formulae take the form of Chebyshev polynomials, whose coefficients are listed in the ephemeris data files.
In order to extract planetary positions from the DE430 files, I wrote the tool ephemerisCompute, which computes the right ascension and declination of each planet from the DE430 data. It can also compute the positions of asteroids and comets, using their orbital elements. If you are interested in learning more about the data format used by DE430, then the comments in this source-code file may help you.