I plan to fly into Madrid on New Year's, 2014. Will I be able to take a train that same day to Seville?
Yes, trains run on a reduced schedule. New Year's Day is pretty dead in Spain.
Yes, make sure you are aware of holiday schedules. We were in Spain last year/early this year from the dates of 12/26-1/11. We were amazed by how many close calls we encountered; one episode, thinking the bus would take us to Malaga from Ronda at 10am, getting there a bit early at 9am, only to see that they were on holiday schedule on NYE. Normal schedule had a bus to Malaga frequenly, maybe hourly. if we did not catch that bus, we would have had to wait 10 hours to catch an evening bus; our problem was that we were meeting a mountain bike tour group at Malaga airport to be transported to the Sierra nEvada mountains and had to be there by a specific time for the 3 hours ride to rural Sierra Nevada spain. If we missed that bus, we would have been screwed. I did not think that they would be on holiday schedule because the holiday was the next day. But I also learned that during the Christmas and New Years time, there are many holy/saint days that are considered holidays. Beware. Do your research. If you want a discounted ticket from Madrid to Sevilla, you will buy your tickets early and you will see what time options you have. That way there will be minimal holiday surprises.