Landscape photography in and around Leipzig is challenging, as the Leipzig area is basically flat, "at the southerly end of the north german plain" as Wikipedia puts it.

For a view, either climb Völkerschlachtdenkmal or the top of City-Hochhaus Leipzig (do some further internet research for those).
There are artificial lakes around leipzig, the leftovers of open-cast mining for lignite or brown coal. Also Clara-Zetkin-Park, a quite large one, might offer motifs for the nature photographer. For some cheating with wildlife photography, visit the zoological garden or Leipzig Zoo.

If you like architecture, you will find there a lot of buildings to be photographed. Leipzig was a center of trade, and one can still see this. For example, he Markt with its Altes Rathaus (market place with the old towm hall), or Brühl (a street in the city between Markt and Hauptbahnhof, a hundred years ago the center of the fur trade), or when visiting Hauptbahnhof or central station. For more information, do further research on this.

Drallewatsch has been mentioned. The student scene is around Südplatz (also a tramway station) on Karl-Liebknecht-Strasse.

For some typical flair, have a Gose at Bayrischer Bahnhof, or even better and unique for its old time inn charm at Gosenstube Ohne Bedenken, Menckestrasse 5, Leipzig-Gohlis, Tramway Line 12, Station Fritz-Seger-Str. They have also a quite informative homepage. Gose is a top fermented beer of the family of sour wheat beers, a quite trendy style in the US of A, as I was told.