In Germany, there are two types of post offices: the normal "Post" and the so-called post agency (Postagentur). Post agencies are newsagents or stationary shops with a post office counter inside. They offer most of the usual services. You can identify both types by the yellow "Post" sign outside.
Am Hauptbahnhof 16-18 (near central station)
66111 Saarbrücken
9am-6pm, Sat: 9am-1pm
Dudweilerstr. 17
66111 Saarbrücken
(opposite Johanniskirche, close to the town hall / Rathaus)
9am-6pm, Sat: 9-12am
Lessingstr. 34
66121 Saarbrücken
(opposite Concord supermarket, Mainzer Str.)
8am-6pm, Sat: 8-12:15am
Neugrabenweg 2b
66123 Saarbrücken
(relatively close to Am Homburg)
9am-1pm, 2pm-4:30pm, Sat: 9-11am
Saarbrücker Str. 292-294
66125 Saarbrücken
(Dudweiler)
9-12am, 2-5pm, Sat: 9-12am
Scheidter Str. 260
66125 Saarbrücken (Dudweiler)
9am-1pm, 3-6pm, Sat: 9-12am
If you need a postcode in Germany, you can find it online.
How to address a letter in Germany (pdf-file)
When moving, you may want the post to redirect your mail (Nachsendeauftrag) or store it for you for a certain period of time. There are different services available (e.g. 6 months redirection: around 15 Euros).