Have you ever traveled near your home?
I live in Busan, but sometimes I change the mood and travel within Busan. I went there last week, so I'll recommend you where and what to do for 2 days and 1 night in Busan.
The best time to travel is Friday and Saturday including weekend Saturday or Saturday and Sunday. It may be crowded, but you can enjoy various events in Busan on Saturdays.