The New Town is located between the Old Town (in the south), Park Traugutta (in the north) and Plac Krasinkich (in the west). If you are coming from the Old Town you might want to continue walking to Krasinski Palace and Park, to see the sights there. Afterwards you can easily catch the metro, bus or tram at Ratusz Arsenal.
If you want to make the New Town your starting point you will to have to walk around 10-15 minutes from the bus stop in Bonifraterska Street (bus lines: 100, 116, 122, 127, 174, 175, 195, 303), then take Franiszkanska Street to get to the New Town. But remember the route through Krasinski Park is more beautiful, but it takes 5 minutes longer.