This year is VERY special as Chuseok comes early and is surrounded by two other national holidays. This means that Tuesday 10/3 all the way through Monday 10/9 no work and it's all paid time off!
For Chuseok, tourist places are busy as visitors and residents alike both go. That being said, I have lived here for six years and am still amazed at how quiet other areas are during the holiday. Some neighborhoods, like Gangnam are pin drop quiet.
The holiday is very much time to spend with family, so the night before and day of are extremely quiet. As the weekend nears, most come back to Seoul, with Saturday night at almost normal levels.
Transit usually runs later to accommodate holiday travelers. As for taking a train or a bus to another city it is VERY difficult. Roads are slam full, meaning bus journeys can take double the scheduled time. Tickets are usually available for buses though. Train tickets, however, are much more difficult to come by as these are only sold at certain periods of time through a special site in Korean. Normally, you can book the English site tickets one month in advance, but at this time, you can't. You have to wait for the special sale to end and then choose from what is left over.
All this aside, as a tourist, you should have a normal Seoul experience. As for the museums, they should be open, but with the long holiday, check the schedule as they may shift their one closing day around. For food and nightlife, in busy areas some places will be open but not all. It will also feel empty in those areas compared to usual.