When you get to tegel, find the public transport office - its by the bus stops. They can sell you day tickets. They will also give you a public transport map of berlin - free
On other days you can buy an all day ticket at machines. They have a credit card keypad but I have always paid cash. You can change the display to be in English.
Get a Euro pre-payment card. I have a fairfx one. I can top it up online, and pull cash from ATM's. cheaper than using your UK debit card. I suppose I can use it in shops etc but never have.
https://www.berlin.de/en/public-transportation/ is good too
Wear good shoes. lots of stairs to climb up and down. Few stations have escalators, so you will be up and down stairs a lot!.
buses are quite rare. they only seem to fill in where there is no s-bahn, U-bahn or tram.
Be careful in some parts of the city at night. Neukoln especially. But people have been attacked on public transport.
See how they are getting on with the new U-Bahn route from brandenburg gate to Alexanderplatz U55/U5 link
Get an S-bahn train from zoo to Warschauer strasse. There are some nice little stations on there. My favourite is Hackescher markt but Bellevue is nice too
go to Warschauer strasse U-bahn. Its an overhead railway. Again some nice stations, and a nice rooftop view. Plunges underground I think after Nollendorfplatz
Eberswalder strasse is nice. U-bahn trains change from underground to elevated there. Lots of trams there too. get one to Nordbanhof - it passes a long stretch of restored Berlin Wall
Tram trips through N berlin are interesting. Lots of DDR tenements.
Take the train to Brandenburg. Nice small town. Double decker loco push pull. Aircon if its hot!
Non train stuff
Do a river trip (They go from near Hautbanhof). Note the pock marks in the canalised river banks. the fighting at the end of WW2 was savage.
Go to Treptower Park. Its on the S-Bahn. Look for the russian WW2 cemetary/memorial. Thought provoking communist architecture.
Get a S-Bahn to Tiergarten. Walk through Tiergarten park. There is a small cafe by the big Prussian memorial in case you need beer/coke/coffee etc half way. When you get to Brandenburg Gate (Brandenburger Tor {Tor = gate} ) look for the row of cobbles in the street where the wall used to run. You can now just walk over it now but for much of my lifetime you couldnt. Have wurst and chips in Unter den Linden because you can now, but for a long time you couldn't.
The row of cobbles follows much of the route of the wall, with a plaque saying Beliner Mauer every so often. Once i walked from Ostbanhof to Postdammerplatz along the wall route. I found the memorial to Peter Fechter in Zimmerstrasse. he was shot in the pelvis and left to bleed out screaming in agony. 17th August 1962
if you get off the beaten track, you may find small plaques set in the pavement by doorways to blocks of flats. names of people and dates. They are the names of people who disappeared during WW2. Google stolperstein (trip-stone)
sorry to bring up some of the disturbing history of Berlin, but the divided city fascinated me. I have family there now so I have been taken to places off the usual tourist trail.
ß means double s. Strasse can also be spelled straße.
Hope helps
. S-Nordbahnhof has a number of photographs of time behind the Berlin Wall and how the U-bahn & S-bahn coped. There's a memorial to the transported jews at S-Grunewald
