I prefer local / cultural / old towns, and authentic experiences as opposed to "westernized" cities.
I'm afraid that all the cities you plan to visit are westernized to some degree. All have large shopping malls for example, just as you would find in London or Paris. You would have to go off the beaten track a bit to avoid this. Smaller towns such as Toruń and Zamość are well worth a visit, but English will not be as widely spoken in the smaller towns as it is in the cities.
I heard there are forests to the east. Would love to see that and any other recommended sceneray
There are indeed. Białystok would be a starting point for visiting Białowieża forest, a UNESCO world heritage site ( the one which has been in the news often as unfortunately extensive logging has been allowed to take place there ). Here are a couple of links so you can read about it:-
I would love to go myself but I think it's probably a better idea to visit in summer. Poland can be very cold in the winter.
In the South East is Bieszczady National Park which I have been to and is well worth a visit:-
If you want to see snow then Zakopane is the place as others here have suggested being a ski resort, but it is VERY touristy. Think wooden chalet style shops, although they are mixed in with original Goral wooden buildings typical of the region. If you have a clear day you will be able to see the Tatra mountains. Despite it being touristy I am glad I went to see it, and I'm sure you would enjoy it. I went by car but I'm pretty sure there is a bus from Kraków which takes about 1.5 hours.
Do you think this is a bit ambitious?
Go for it! You will be fine. Although I have stayed with friends in Poland, I've also gone on my own many times. LIke you, the last thing I would want to do is go on a group type holiday. Plan carefully, book e.g train tickets in advance if you can as this will save time and queues, and most importantly, have fun!!