Busan is easy to navigate by the metro. The sights are spread apart, so no particular area is better an the others as far as access to the tourist spots.
For example the ferry terminal and Jagalchi are in Nampo-dong. Beomeosa is in the north near Nopo-dong. Yonggungsa is on the east side of Haeundae Beach.
I typically stay in Haeundae. You are always within walking distance of a metro, there's food options and some nightlife.
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