Mumbai (= Bombay) is well-connected to many cities around the world by direct flights and to most cities in India by flights, trains, and roadways. Please note that most international visitors will require a visa for India (just as Indian citizens require a visa to visit their country), and also that December-January tends to be a busy period in terms of travel to and from India, especially from Europe and North America; thus, it is advisable to book early. Here are some useful links.