Hi,
When I am in Berlin, as I will be in June 2017, I don't stay at Pension Peters but in the same area, Berlin-Charlottenburg. On the various districts in Berlin, that is well recommended area, nice area around Kantstrasse, Savignyplatz, and Kurfürstendamm, numerous restaurants of various cuisines, book stores, cafes, small hotels, Pensionen. A very nice area to stroll. I always stay in that area, convenient too.
Regarding the public transport, from Peters (at the corner of Kantstrasse and Savignyplatz), the closest S-Bahn station is "Savignyplatz" station, where you can catch the S-Bahn to Potsdam, Wannsee, etc and eastwards to Berlin Hbf.
Berlin-Lichtenberg has a train station, just as Charlottenburg has Bahnhof Zoo. I was there once, in 2003 coming back from Gdansk to change trains to Bahnhof Zoo, ie, before Berlin Hbf was completed. it must be all refurbished since then. I don't know the area recently but the standard price for a DZ (double) in Berlin will be ca 65-95 Euro. I pay 40 Euro for a EZ (single), just to give you an indication prices in Berlin. More to the east such as the districts of Karlshorst or Köpenick (change at Ostkreuz station) might fit your bill. The last time I was in Karlshorst was this past June, a lot has changed since I saw it over 16 years ago. Whether you pick someplace in the west or east, they are both well served by the subway (U-Bahn and S-Bahn).
Have you looked at Berlin-Alexanderplatz, that area? Very lively, lots going on, a junction point linking all different parts of the city by S-Bahn.