What is the best way to get to Taiyuan South Train Station from either airport or the city itself?

You can take bus no.201, get off at Taiyu Road Rongjun North Street. Then walk northwest around 0.6 miles (1km) to find the South Railway Station. It takes around half an hour.
From downtown area, public bus no.502, 824, 868 or 909 can take you to the South Train Station Bus Stop. Then just walk to the ticket hall directly.
In addition, you can take shuttle bus line 3 to get to South Railway Station directly. The operating hours are from 09:30 to 21:30 with the interval of 20 minutes. The duration is about 20 mins. And the ticket fare is CNY 13 per person.