Bydgoszcz public transportation system consists of buses and trams. To get your monthly or semester ticket, you need to have a City Transportation Card, which is called Bydgoska Karta Miejska.
Your student card can work as City Transportation Card. Once you activate your student card, you can encode a ticket on it. To encode a ticket you just need to use a vending machine or pass your card to the sales assistant at a kiosk.
Instructions how to get City Transportation Card you can find at the website: bydgoskakartamiejska.com.pl (you can choose English or German version). After completing and sending the form you will receive a confirmation email with a number of application. Take it and your Polish student card and go to ul. Toruńska 174a, COK (Customer Service Centre) at the seat of the Municipal Roads and Public Transport Authority, from Monday to Friday from 7.00 to 17.00.
You will have your student card activated there as a transportation card. Then to buy tickets online you have to create an account: https://bydgoskakartamiejska.com.pl/en/register-page. Here you have a long lists of points where you can buy tickets either single (jednorazowy) or monthly (okresowy): https://bydgoskakartamiejska.com.pl/en/points-of-sale
Single student ticket costs – 1,90 zlotys. Monthly student ticket (daily and night lines) – 56 zlotys. With payment card you can buy a ticket on most buses and trams. Always have a student card with you – it entitles you to a discount. You are also entitled to discounts with ISIC (International Student Identity Card) or EURO 26 Student – age limit 26.
Another useful site to plan your trips is called "jak dojadę" which also works in English.