The closest international airports are Václav Havel airport (PRG) in Prague, located 170 km to the north or a smaller airport in Linz, Austria (LNZ), 110 km south from České Budějovice. Both are connected with České Budějovice via direct train or bus lines. You may also opt to fly to Munich, Vienna or Brno but it will require a more complicated transport from there.
Short and very helpful tutorial video here
Direct trains from Hlavni nadrazi (Main Railway Station) in Prague go roughly every hour to České Budějovice. There are two types of train - rychlík (R) and expres (Ex), with the latter being faster and more comfortable. The trip takes around 2-2.5 hours and costs 170-200 CZK (€7-8) for the second class. You can check the timetable and buy tickets at the station, online or via an official app. There are direct trains between Linz and České Budějovice as well.
There's plenty of buses going down south as well but they all leave from different locations around Prague so it might be a bit confusing. Yellow buses are very popular as they are very comfortable, have wifi onboard, provide free hot drink and newspapers/magazines and have a hostess onboard who speaks English. They depart from Na Knizeci bus station on the metro, line B (stop Andel). It's advisable to buy tickets online or through the official app, as they sell out quickly.
Flixbus also operates direct buses from Prague, Linz and other cities.
The more expensive but convenient way is to order a private shuttle directly from anywhere in Prague or even Linz, Vienna or Salzburg. It needs to be ordered online beforehand and costs 1000 CZK (€40) per person (minimum of 2 people required per transport).
You will find a comprehensive guide on how to travel around České Budějovice in the venue section.