For a real insight into contemporary Leipzig, I recommend a visit to the G2 Kunsthalle, which houses the Hildebrandt collection. It's on Gottschedstraße, across the ring road from the Stasi headquarters. It has a fascinating collection of contemporary art, much of it from the so-called New Leipzig School, as well as changing exhibitions. It's open on Wednesdays from 3pm to 8pm, but you can visit at other times by booking an English or German tour. Tours can be booked online on the gallery web site - g2-leipzig.de/en.
If you have time at the end of your walk, I recommend a trip to Café Grundmann on August-Bebel-Straße, an Art Deco coffee house in the arty Südvorstadt area of town. You can get there on a number 11 tram from Augustusplatz. Get off at Südplatz and walk down Schenkendorfstraße to August-Bebel-Straße. In Café Grundmann, you can enjoy delicious German food and wines. I particularly encourage you to try some German reds.
You can walk this route at any time of the day or night.
After dark, you'll feel the city's night-time buzz and perhaps be able to visualise the pro-democracy demonstrations that took place on Monday evenings during the Leipzig autumn of 1989, but you won't be able to stop off at the museums.
During the day you'll get a better view of the architecture and be able to visit museums and galleries that interest you. Avoid Mondays as most museums are closed then, though the Museum in the Runde Ecke is open Monday to Sunday from 10am to 6pm. The Museum der bildenden Künste (museum of fine art) is open Tuesday and Thursday to Sunday 10am to 6pm. On Wednesdays, it's open 12pm to 8pm. The Zeitgeschichtliches Form (contemporary history museum) is open 9am to 6pmTuesday to Friday and 10am to 6pm at the weekend. On Sundays, you can attend a service at the Thomaskirche. These take place at 9.30am and 6pm. If you're lucky, you might hear the St Thomas Boys Choir. If you want to be sure of hearing the choir, check the church's web site in advance - www.thomaskirche.org.
Leipzig is generally a pretty safe city, and people are usually friendly. It can get quite hot and sunny in summer, so take some sunscreen and a bottle of water. In winter, it can get really cold, so wrap up warm.