Town outside Berlin

Potsdam (southwest of Berlin, 170,000 inhabitants): Frequent trains to Berlin (RE [Regional Express] fast) and S7 (Suburban Train).
Werder (a little further away from Berlin than Potsdam, 25,000): RE to Berlin.
Oranienburg (north of Berlin, 45,000): RE and S1.
Bernau (northeast,37,000): RE and S2.
Köpenick (southeast, belongs to Berlin,64,000): RE and S3.
https://www.bahn.com Hbf=Hauptbahnhof=main station, Bf=Bahnhof=station
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I forgot: Werder (Havel), because there are several Werder.

Keep in mind that Berlin is huge. If you travel for 30 minutes you are still in the city, although some of the outskirts can feel very rural. Some of Berlins suburbs are actually former villages, and sometimes haven't changed much.

I would recommend Potsdam, which is a larger city, but with some very nice quarters. To the southeast of Babelberg park, one of the royal parks of Potsdam, is the district of Babelsberg. The part of the district closest to the park is called Nowawes, a former village with cute little houses from the 18th century built for religious refugees from Bohemia. Some of these houses were turned into holiday homes, in case you don't want to stay in a hotel. Parking there is free, the next train station to Berlin is a 5-10 min walk, and in the evening you can enjoy one of the poshest parts of Berlin/Potsdam, with two royal parks and several lakes.

Potsdam would be a good location.

In September, 2016, I enjoyed my stay at "Deutsches Haus", Restaurant & Pension, Dallgow-Döberitz. This pension and restaurant is located adjacent to the Dallgow-Döberitz Bahnhof (railway station), so travel into downtown Berlin is a breeze. The rooms at this small hotel are fine. Included at this property is a great restaurant + bar. Wifi (W-LAN) and secure parking are included at no additional charge. Recommended.

"Deutsches Haus": https://www.deutsches-haus-dallgow.de/

This place used to be a suburb, like Köpenick, but the opposite direction...Spandau. Babelsberg next to Potsdam is all right. You could try Kleinmachnow, also near Potsdam.

Thank you guys. I love this RS forum and the people on it.