Gatwick to Tower Bridge, I would take the Thameslink train to Blackfriars station, then follow signs to the Circle or District line eastbound, alight at the fourth stop, Tower Hill station.
For your info, Blackfriars Rail station platforms span the river Thames north/south, the platform has a glass wall and the views eastwards down the river with city scape and landmarks are excellent, you know your in London, although the train from Gatwick might arrive on the west facing side.
Gatwick was shoe-horned into the Oyster fare boundary, so fares are varied and complex, You can use Oyster/contactless bank card/ Apple/android/phonepay, there is also a One Day National Rail Travelcard, a combo paper ticket that includes the Gatwick round trip and all day unlimited use in London of underground/buses/DLR