By air. Copenhagen Airport (CPH) at Kastrup is the main hub, about 8km south of the city centre and well connected by Metro (around 15 minutes to the centre). Most European capitals have multiple daily flights. From London there are frequent services with SAS, British Airways, and Ryanair among others. Billund Airport in central Jutland handles a good number of European routes and is more convenient if you're heading anywhere other than Copenhagen.
By train. Direct trains run from Hamburg to Copenhagen in about four and a half hours, crossing into Denmark via the Jutland line or by ferry connection. Eurostar connections from London require a change (typically via Brussels or Amsterdam), but the journey is doable in a day. DSB (Danske Statsbaner) is the national rail operator and runs frequent intercity services between Copenhagen, Odense, and Aarhus. Rail passes covering Scandinavia are available and worth considering if you plan to travel widely.
By ferry. Ferry connections are still a practical way to arrive. DFDS runs routes from Oslo to Copenhagen and from various German and Dutch ports. From Germany, the Puttgarden–Rødby crossing on the E47 corridor takes about 45 minutes and runs very frequently — convenient if you're driving through from Central Europe. Ferries from Norway (Kristiansand, Bergen) arrive at Hirtshals in northern Jutland.
By road. Denmark is connected by motorway to Germany. From Sweden, the Øresund Bridge links Malmö directly to Copenhagen by road and rail — one of the more satisfying ways to arrive, with good views of the bridge and the city skyline.