The sole official language is Indonesian, known as Bahasa Indonesia. Indonesian adopted a number of loan words from Arabic, Dutch, and Sanskrit.
It is similar to Malay (spoken in Malaysia, Brunei, and Singapore) and speakers of both languages can comprehend each other to a large extent.
The main differences are in the loan words: Malay borrowed mainly from English, while Indonesia borrowed mainly from Dutch.
Written phonetically with the Latin alphabet and with a fairly logical grammar, Indonesian is generally regarded as one of the easiest languages to learn, and A.M. Almatsier's The Easy Way to Master the Indonesian Language, a 200 page small paperback, is an excellent starting point. It can be found in any Indonesian bookstore for less than 3 dollars.
The language went through a series of spelling reforms in the 1950s and 60s to smoothe over differences with Malay and expunge its Dutch roots. Although the reforms are long complete, you may still see old signs with dj for j, j for y, or oe for u.
Many educated Indonesians understand and are able to speak English. While Indonesian is the lingua franca throughout the archipelago, there are thousands of local languages as well, and if you really get off the beaten track you may have to learn them as well. Some ethnic Chinese communities continue to speak various Chinese dialects, most notably Hokkien in Medan and Teochew in Pontianak.
Most educated seniors (70 years/older) in Indonesia understand Dutch, but realistically speaking English is far more useful these days. Many educated Muslims, especially those who graduated from Islamic religious institutes, understand Arabic to varying degrees.
English language TV channels are available on most hotels. MetroTV (local TV channel) broadcasts news in Chinese from Monday to Friday at 07.00 AM. MetroTV also broacasts news in English from Monday to Friday at 07.30 AM. TVRI (state owned TV station) broadcasts news in English from Monday to Friday at 04.30 PM in the afternoon. All schedules are in Waktu Indonesia Barat (WIB), which is 7 hours ahead of Greenwich Mean Time and includes the capital city of Jakarta.