This is probably the most important part of every trip. In order not to ruin your first day and to make it to the return flight, you have to deal with public transport. You have accommodation in the old town, i.e. in the center of Gdańsk.
"Gdańsk Główny" translates to English as "Main Gdańsk / Central Gdańsk", there is also Gdańsk Śródmieście, translated to English the same way and these are two different stations.
Your destination station is "Gdańsk Główny". That is the penultimate (last-but-one) train station. If you miss it and get off by mistake at the last station "Gdańsk Śródmieście", nothing is lost, in fact you can walk from the "Śródmieście" station to the main Gdańsk in 15 - 20 minutes. But remember to buy a ticket to the "Gdańsk Główny" station on the train.