The Meenakshi Amman Temple is a must-visit when in Madurai, especially if you're interested in South Indian temple architecture and history. The temple is dedicated to the goddess Meenakshi and is a popular pilgrimage site. The temple complex is vast and includes several smaller temples and shrines, as well as a 1000-pillar hall and a tank. Make sure to dress appropriately, as the temple authorities enforce a dress code.
The Thirumalai Nayak Palace was built in the 17th century by King Thirumalai Nayak and is a fine example of the Indo-Saracenic style of architecture. The palace was once much larger than it is now, but what remains is still impressive and warrants a visit. The palace houses several halls, including the main audience hall, dance hall, and royal bedchamber. The light and sound show in the evening is also a popular attraction.
If you have more time, consider visiting the Gandhi Memorial Museum or the Koodal Azhagar Temple. If you have less time, skip the Thirumalai Nayak Palace and instead visit the Vandiyur Mariamman Teppakulam, a large tank and popular picnic spot.