Seollal(Lunar New Year) holiday is usually 3-5 days long. While some places may be closed for part of the holiday, most places popular among international travelers will continue their business.
Theme parks such as Everland, Lotte World, Nami Island, Petite France, Trickeye Museum, etc. are usually open 365 days a year, no holiday. Palaces only close once a week, regardless of Seollal so please check the date. (e.g. Gyeongbok Palace closes on Tuesdays, Changdeok Palace closes on Mondays)
Some places may be closed for part of the holiday, most places popular among international travelers will be open. If you have a particular restaurant you want to go to, please try to contact them in advance to check.
About 32 million people visit their hometown during New Year holiday. (That's more about 60% of Korean's total population) Among these 32 million, 80% drive their car. Imagine how many cars will be on the road. Most routes will take 2-3 times the normal traveling hours. (Seoul -> Busan would normally take 5 hours but on holidays, it can take up to 15+ hours)
We recommend you to stay in Seoul during the holidays and especially avoid traveling outside on the first and last day of the period. On the contrary to traffic on the road, Seoul will be pleasantly empty.
Traveling to places nearby such as Nami Island and Everland will not be too bad, but still, be prepared for traffic.
Bus and train tickets to travel far are next to impossible to get. There is a special period for reserving tickets for the holiday and all options will be sold out within 1 minute of opening.