The highlight of the town of Aranjuez and no doubt the motivation behind the majority of visits is the Royal Palace, a sprawling complex of ornate hallways, ballrooms, and absolutely stunning gardens. It was built by King Phillip II in the 17th century. You can get there easily on Cercanias (line C-3) from the Atocha, Sol, and Nuevos Ministerios stations in downtown Madrid. If you're going to visit on the weekend or a holiday, I'd recommend getting there bright and early as the access is somewhat slow and the lines can get lengthy quite fast.
The best parts of the interior are due to renovations by Queen Isabel II. My favorite parts were the Arabian Study, a truly overwhelming smoking room that's reminiscent of the most ornate parts of the Alhambra, but while the latter's ornate cupolas are only relics of what they once were, the Arabian Study is a technicolor psychedelic room which invites you to spend an hour contemplating the details (and that's taking into account that it's only large enough to fit 5-10 people. The other highlight was the Porcelain Study, a room whose walls are covered in porcelain vines and oriental motifs typical of Spanish Rococo design. These two rooms alone make the trip worth it, but the various antechambers, dancing rooms, chapels, and throne rooms are also wonderful in their own right.