Madurai second largest city in TN, can be reach best by train from Chennai. Catch Pandian Exp, Nellai Exp you ll reach by 6-6.30.
Checkin to any hotel of your budget just come out of station and cross the road you ll find plenty of Hotels and touts alsowho can help you to find rooms.
My suggestion is checkin in Hotel, be fresh and go to Meenakshi Temple in morning itself. This temple is very big and 2 hrs atleast you need to cover the entire Temple Site.Temple has 4 Main gopurams(Gates), East gate remain closed. Tourist takes West Gate generaly as it is easily accessible but it is very crowed so try North Gate which is less crowed and local people use that gate only.
If you are new , engage a guide Rs 50 (max) he will show you the Temple. If you have camera please buy a camera tickets(Rs 50).
After coming out of Temple if you you want, you can visit saree shops surrounding the temple.Madurai Cotton is famous but be careful sometime you can be cheated.
After coming out of Meenakshi temple go to any Tour Operator and book seat for local sight seeing (max Rs 120 pp). Tour will start around 3.30 pm. They will cover many more temples , Nayakar Palace and Gandhi Museum and drop the people near Meenakshi Temple around 7.30-8.00 nite to visit the temple. Since temple closed by 9.30 pm it is better cover the temple in morning itself.
If you want you can go to Meenakshi Temple in evening also to see aarti and lightings.
Madurai is a place which connects many temple city and places in TN and Kerala.
You can go to Rameswaram , Kanyakumari, Trichy, Kodaikanal, Courtallam, Thekkady and Munnar from there. You can book package tour or hire a cab. To hire a cab please dont go to Tour Operators they will charge you more better go to Taxi stand which is near Railway station and negotiate with them directly. Language is not a barrier there people speak hindi and english both.