There are 3 main airports - Sofia, Varna and Bourgas. Sofia airport is operating all year round and serves many regular destinations to/from most European big airports. Varna and Burgas airports are operating mostly from May to October because of the big touristic flow to Black Sea resorts.
Wizzair offers low-cost flights from most European cities to Bulgaria if you book in advance.
There are several international railways destinations to/from Budapest (Hungary), Bucarest (Romania), Belgrade (Serbia), Istanbul (Turkey), Thessaloniki (Greece), Alexandropolis (Greece), Moskow (Russia), Minsk (Belarus)
Timetable of Bulgarian State Railways
There is a well developed net of bus lines to/from most European cities - ask your local travel agencies. This is the cheapest mean of transport and is faster than the train (for example the 1300 km journey from Sofia to Prague takes about 20 hours)
Bulgaria is on the crossroad between Europe, Asia and Africa and there are many roads passing through: - E80 from Europe to Istanbul (Turkey); - E79 from Europe to Thesaloniki (Greece); E85, E87, E871 ...