Taiyuan's coffee culture has grown substantially since 2015, and the city now has a decent number of independent cafés alongside the national chains (Luckin, Manner, and the rest). The best concentration of independent coffee shops is in the streets near the Shanxi Museum and along the river embankment on the western side of the city — good places for a quiet morning with a flat white before heading to the museum or out to Jinci Temple.
The cafés in the Jinci area itself are mainly tourist-facing and unremarkable, but the walk along the stream approaching the temple in the early morning, with tea rather than coffee from a roadside vendor, is pleasant. For anything more serious, the city-centre cafés are better.