There is an international airport in Warsaw: [http://www.lotnisko-chopina.pl/?lang=en Warsaw Frederic Chopin Airport]: For those travelling from Warsaw serviced by [http://www.lotnisko-chopina.pl/katalog/dojazd/en/dojazd_autobus_express.php Polski Express]. See also: [http://polskiexpress.pl/rozklad Routes/Timetable]
The Polish State Railways have train connections from Warsaw via Bialystok. The train takes two and a half hours and runs every two hours. It is also possible to take a train to Hajnowka which is closer to the forest, but the trains are slower and less frequent. The forest can be reached by both places by bus.
See: [http://www.pkp.pl/ Polish State Railways]
You can easily get there by car from Warsaw via Bialystok.
There are bus-services from Bia?ystok and Hajnowka to Bialowieza. There are several busses running per day, but they are run by different companies it's hard to get a timetable of all of them.