We have booked an apartment in Bornholmer Straße. There are 4 of us. Should we get a travel pass, just pay for each bus ride of would it be just as cheap to take a taxi to places like museum Island and Tiergarten?
The Verkehrsverbund Berlin-Brandenburg is the metro district for the Länder (states) of Berlin and Brandenburg. The English version of their website is www.vbb.de/en/index.html.
A map of the S- & U-Bahn lines showing stops and the zones is here.
The page showing all of the fares is here. For the inner two zones (AB) a daily ticket is 6,70€ adult, 16,20€ small group, up to 5.
For round trip travel from the Bornholmer Str. S-Bahn station to the Tiergarten S-Bahn station, for 4 adults, the website recommends the Berlin AB small group daily ticket for 16,20€. The 21 minute route they show uses 2 S-Bahn (S25 to S75) with a 5 minute change at Friedrichstr. Bhf. When I tried to limit the conveyances to buses, it said there was no way to do it. The Bahn website also does not show a bus connection.
I can imagine that many S-Bahn stations will not have a manned ticket counter, only a ticket machine. The Bahn website does not show a ticket counter at Bornholmerstr Bhf. If you can't figure out the machine you might have to take a taxi.
You can buy a 7-day card for € 28,80 (Berlin zones A+B, which will be enough for all travel to touristic sites except Postdam and Sachsenhausen, for which you can buy supplements for the days you need).
More info: http://www.bvg.de/index.php/en/1236666/name/Weekly%2B+Monthly+and+Annual+Tickets/article/77395.html
You can buy a ticket to Szczecin (German name: Stettin). That's only 10 Euro and 2 hours travel from Berlin Hauptbahnhof or Berlin Gesundbrunnen Station (more trains) - direct connections or with a change in Angermünde, that takes usually about 5 minutes (usually at the same platform)