Bielsko-Biała is a city in Poland.
North towards Katowice
Near the train station there is very popular hitching place at a bus stop. Another possibility is the petrol station close that bus stop.
East towards Cieszyn, Brno (Czech Republic)
The express roads cross just near Tesco. To get there just take any bus from the train station. There is also a free bus service for Tesco customers. You can walk up from Tesco to the intersection and hitch from one of the two on-ramps for this direction. Unfortunately, the new highway has been designed so that there is nowhere safe for a car to pull over, so it may take a long time of waiting before someone decides to risk stopping for you.
This road enters the Czech Republic at the Chotěbuz-Cieszyn-Boguszowice border crossing.
East towards Kraków (N52)
It is possible to hichhike close to Tesco but easier is to go from train station with bus #6 to the final stop Lipnik, or walk from the centre along the street 11-Listopad to the end and hichhike from start of Krakowska street. It is not allowed but common.
Take the bus #2 from the train station in the direction Mikuszowice-Stalownik. At the last stop or the second-last, you can hitchhike from the bus stop.