The main reason to travel beyond Kursk is Prokhorovka, the site of the decisive tank engagement of the 1943 battle. The Prokhorovka memorial complex — about 80 km north near Belgorod — is more evocative than anything in the city itself, set on the actual steppe where the battle was fought. Allow half a day and consider combining it with Belgorod, which has its own WWII museum and a city centre worth a short walk.
The Korennaya Pustin Monastery, about 30 km south of Kursk in the village of Svoboda, is the original home of the Kursk Root Icon. The pilgrimage site has been active since the fourteenth century and includes sacred springs; the setting in a ravine by the Tuskar River is genuinely peaceful.