The primary gateway to Indonesia is the Sukarno-Hatta International Airport of Jakarta (also known as Cengkareng CKG), but there are increasingly more international airlines that offer direct flights to Bali. From Jakarta, Garuda Indonesia and Sempati Air have around a dozen departure times to Bali. Internationally, Garuda also has direct flights from Singapore, Hong Kong, and a number of Australian cities as well as flights from Los Angeles, Amsterdam, and other European cities.
Qantas Airlines of Australia provides direct flights from a number of Australian cities as well as a number of European cities (e.g. London). KLM of the Netherlands was one of the first international airlines to have an international flight to Bali, and it will take you to Bali from Amsterdam and many other cities in Europe.