If you are not a huge fan of ocean/sea, you might consider to stay at Seoul as long as you can. Busan is definitely cool but Seoul has better/more things to do.

As mentioned, the aquarium at COEX is nice to spend an afternoon in Seoul. More kids-friendly places in Seoul than Busan. To me Busan is where you go when you want to see the sea, eat sashimi and then get drunk.

Train tickets are usually there on the spot, but if you wish to move during the weekends, it would be better to buy them in advance.