Your own feet excluded, the cheapest mode of moving around is municipal transportation, i.e. buses, trams, local-range trains and metro. If you are planning on using it only from time to time, then single-fare or short-term tickets are an optimal solution for you. If you use public transportation on regular basis then the best solution is to buy a long-term ticket (the longer the term, the lower the per-day cost). Remember that once you have your student ID card, you will be entitled to a reduced fare (50% of the standard fare: 30-day ticket - starting at 55 PLN, 90-day
starting at 140 PLN). More information on public transportation and
current ticket prices are available on the website of the Warsaw Public Transport Authority.
If you are looking for an alternative to public transport, you may want to check out various bike-sharing systems accessible in Warsaw (though they are typically unavailable in the winter season) or even car or motor-scooter sharing services. You can access them using your own smartphone.