Traveling in Poland

Rome2Rio.com is often a good starting point to answer questions like this. It appears that--unless you want to rent a car--your best bet is to start with the train to Hajnowka (closer to the park) or Bialystok (farther from the park). Those are two completely different train routes, so you'd need to investigate your onward travel before making a decision.

Everything Rome2Rio coughs up seems to end with a taxi ride, but sometimes it turns out that there is public transportation available that Rome2Rio doesn't know about. If you or other members of your party will have some time in Gdansk, I'd make a beeline for the local tourist office and ask for help in deciding how to reach the park. Even if you do have to make part of the trip by taxi, the cost shouldn't be exorbitant split 3 or 4 ways.

I would just rent a car from Gdansk, I think that would give you the most flexibility and really should not be a big deal unless you don't know how to drive a manual (and there are no automatic cars available). All the directions are from Bialystok as a jumping point, not Gdansk. Googling Wikitravel + the place will usually give you good answers. Please report back about your experience! I'm Polish and sadly have not been to this park. I'd love to hear about your trip there.

http://wikitravel.org/en/Białowieża_National_Park
http://www.lonelyplanet.com/poland/mazovia-and-podlasie/bialowieza-national-park/introduction

Bus service from nearest town
http://voyagertrans.pl/en/

Thank you! I appreciate the tips...

The best way is by car. Travelling by public transport can be done but it will take longer.

You can take a direct train from Gdańsk to Białystok that takes nearly 7 hours (8:08-14:50) or go via Warsaw with a connection at Warszawa Wschodnia (do not go to the Central station because you will be backtracking). The bus from Białystok to Białowieża takes 2 hours. The bus station in Białystok is just on the other side of the railway tracks but be aware that Voyager buses do not depart from the bus station itself but from a bus platform numbered 059 (one of those where local municipal MPK buses stop)
trains:
http://rozklad-pkp.pl/en
(Gdańsk Główny- Białystok)
Most direct buses are Voyager linked above but there are also bus connections with a change in Hajnówka or Bielsk Podlaski, should you miss all the direct ones.

I do not really trust non-Polish websites because they are full of wrong info, rome2rio for example sends you via Siedlce which in my opinion is a recipe for getting stranded in the middle of nowhere (Siedlce) with very little transport to Hajnówka, let alone Białowieża. As a foreigner I would suggest sticking to the main routes, via major cities, not provincial little towns.