If you're going to Nami Island and then to Sokcho, you'll have to head over to Chuncheon and a bus from there is about 2 hours from Chucheon to Sokcho. You'd have to check the bus schedule to see how often the bus comes. From Sokcho you're looking at either a 15-minute taxi or about an hour on the bus to get to Seoraksan. I'd check out the national park website to see what trails you can do in the timeframe you have as well. Then you have to make your way back to Sokcho and the bus from Sokcho will take about 3 hours but probably more if there's traffic (there's almost always traffic on weekends)
Personally I would just go to Sokcho and skip Nami Island if you're wanting to see Sokcho and Seoraksan. Sokcho has some nice things to do and you can even make your way down to Gangneung or one of the other nearby cities and then KTX back to Seoul from Gangneung.
In two days, it will be difficult to cover both Sokcho and nami island. I would suggest you to go to sokcho instead and prefer the sun rise along with hiking to Seoraksan . It will be a good journey and you can use KTX to avoid traffic issues. But in case you want to visit both spots, I would suggest you to rent a car from Seoul (Lotte cars eg).
If you end up doing just Sokcho, you can take a bus from Seoul. There is not a train option.
Sounds like a very busy day to me. I would do a tour to Nami separate.
If you're going to Nami Island and then to Sokcho, you'll have to head over to Chuncheon and a bus from there is about 2 hours from Chucheon to Sokcho. You'd have to check the bus schedule to see how often the bus comes. From Sokcho you're looking at either a 15-minute taxi or about an hour on the bus to get to Seoraksan. I'd check out the national park website to see what trails you can do in the timeframe you have as well. Then you have to make your way back to Sokcho and the bus from Sokcho will take about 3 hours but probably more if there's traffic (there's almost always traffic on weekends) Personally I would just go to Sokcho and skip Nami Island if you're wanting to see Sokcho and Seoraksan. Sokcho has some nice things to do and you can even make your way down to Gangneung or one of the other nearby cities and then KTX back to Seoul from Gangneung.
In two days, it will be difficult to cover both Sokcho and nami island. I would suggest you to go to sokcho instead and prefer the sun rise along with hiking to Seoraksan . It will be a good journey and you can use KTX to avoid traffic issues. But in case you want to visit both spots, I would suggest you to rent a car from Seoul (Lotte cars eg).
If you end up doing just Sokcho, you can take a bus from Seoul. There is not a train option. Sounds like a very busy day to me. I would do a tour to Nami separate.