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.