The historical centre of Bydgoszcz is tiny and you'll rarely need more than a few minutes to reach your destination by foot, but getting to and from the train station, 2km away, is quickest by public transport or taxi, as are trips to the Myślęcinek recreational area, the Exploseum, the Valley of Death, and other destinations outside the city centre.
A total of 46 bus and 10 tram routes plough the streets of Bydgoszcz, of which the latter are by far the most convenient. Tickets can be bought from ticket machines (onboard some buses/trams and scattered around the city), newspaper kiosks, or - rarely - straight from the driver. The cost is 3zł for a single-journey ticket (no changing buses/trams), 4.20zł for a one hour ticket (unlimited changes), and 12zł for a day pass. Note that there are no concessions for foreigners, and tickets must be validated in a special machine at the start of your journey, otherwise you might get caught by a plain-clothes inspector and forced to pay a 258zł fine. To figure out how to best reach your destination, use the invaluably helpful website bydgoszcz.jakdojade.pl, which figures out all connections for you (it also comes in app form).
The most useful and trustworthy taxi company at the moment appears to be MPT Łuczniczka and you should keep an eye out for them - you'll definitely see them standing outside the airport if you fly into Bydgoszcz. There are also taxis sitting just outside of the train station to the right, but we can’t vouch that all of them will be honest at the sound of a foreign accent. Always make sure the taxi you're taking has the official fares displayed somewhere visible; they start at around 7zł for the first kilometre, with a further tariff of about 2.20zł/km after that. Most taxi drivers are honest, although it's unlikely you'll find one who speaks English - having your destination written down can be very useful. Typical prices are 25zł for the journey from the airport to the Stary Rynek (Market Square) and 15-17zł from the train station to the Stary Rynek. At night, as well as on Sundays, prices climb by some 50%.