The city centre is small enough to walk. From the ferry terminal to Kärnan at the top of the ridge is about ten minutes on foot, and most of what you want to see falls between those two points. Sofiero Palace, five kilometres north along the coast, is the main destination that warrants a bus or bike.
City buses run frequently and cover the suburbs, including routes north along the coast to Sofiero. Single tickets can be bought on the Skånetrafiken app or with a card on board. The whole municipality is covered by Skånetrafiken, the regional transport authority — the same tickets work on trains to Malmö and Gothenburg.
Cycling is easy on the flat waterfront and the coast road north. Bike hire is available near the central station.
The ferry to Helsingør is operated by ForSea and runs throughout the day and into the night. Foot passengers pay a small fee; there is no need to book — you simply board the next departure.