By air: Varna Airport (VAR) is about 8 km west of the centre. In summer it receives direct flights from much of Europe — London, Vienna, Warsaw, Amsterdam, and a dozen other cities. Outside the June–September season, connections thin significantly and most European routes disappear. A taxi to the centre takes around 20 minutes; there is also a regular bus connection (line 409).
By rail: The overnight train from Sofia takes around 7–8 hours and is a perfectly civilised way to travel — book a couchette and wake up in Varna. Day trains run too but are slower. The station is central, a short walk from the archaeological museum and the sea garden.
By bus: Express coaches from Sofia cover the distance in around 7 hours and are usually faster and more frequent than the train. Several private operators run the route; the bus terminal is close to the train station. There are also good bus connections to Plovdiv, Burgas, and the resort towns along the coast.
By sea: Varna is a working port. In summer there are ferry connections to Odessa in Ukraine and to Istanbul, though services change from year to year — check current schedules closer to your travel date.