I've been reading around I'm glad I'm going to be in Eastern Poland for the majority of my stay. I heard the Eastern part of the country has less tourist.
I live near Białowieża forest and I can tell you that you've made a good choice if you like trees, as I do.
There's not a great deal to do except walk in the woods, although the standard of hotels in the village has shot up in the last ten years. The Żubrówka hotel is the best (Best Western) and has a great restaurant which serves game, and is very reasonably priced. There's a Russian restaurant called Tsarska nearby, which has sensational food but is expensive by Polish standards. When I have stayed in Białowieża I stayed at the cheaper hotels and ate at the better ones.
You should take a walk along a route called 'Żebra żubra'. This starts near the village and goes through the forest to the Bison Reserve, where you can also see Lynx, Moose, Tarpan and Boar.
Another place most tourists miss (I know several locals who had never heard of it) is 'Miejsce mocy', the 'Place of Power'. This is an ancient religious(?) site with stone circles. It is believed by some to have magical healing powers. I heard a guide say that foresters sometimes observe injured deer coming there for a while, like a hospital. The less romantic fact is that the place has naturally high radiation levels.
A tour of the 'strict reserve' (you have to have a guide) is expensive and, in my opinion, not worth it unless you are a professional botanist or biologist.
If you're coming in winter, try a sleigh ride in the forest. Avoid the hotel-organized tours and contact a guy in Czerlonka who does it for a fraction of the price. He's not hard to find. PM me if you want more info.