Geographically, religiously and even politically Kaveri (Cauvery, in English) is a significant river in South India.

There are many large cities of south india developed on the shores of this river. In some sense Kaveri to the Deccan plain is what Ganges to the Indo-Gangetic plain of the north India.
Kaveri flows through the two Indian states, Karnataka and Tamilnadu, before merging with the Bay of Bengal. Both the farming lands located alone of both the states depends heavily on the river for irrigation. The sharing of water between the two states is a touchy issue, especially when there is scarcity during a drought year.      Though Mysore city is not on the banks of Kaveri, this river has quite a strong relation with Mysore. The river makes a broad arc around the northeastern region of Mysore.
If you are traveling by road or rail from Bangalore to Mysore, you'll cross Kaveri to enter Mysore district. In fact both the road and rail lines pass through Srirangapatna, an island formed by the river.
You can visit many touristic and religious places near Mysore that are located on the banks to Kaveri.
Talakaveri in Kodagu (Coorg) is the origin of this river. Mythologically Kavery is treated as a goddess. So its origin is a very significant religious site.
Talakad where Kaveri takes a sharp turn in its course is a temple town submerged by sand dunes.
KRS Dam (Krishnaraja Sagara) is a massive dam built across Kaveri near Mysore. The more famous tourist spot is the Brindavan Gardens that is located adjacent to the dam. Near by is the Venugopalaswamy Temple , a massive ancient temple translocated from the a submerged village.
Balmuri and Yedamuri Falls are popular picnic spots. Further down in the Ranganthittu Bird Sanctuary. Later the river bifurcates and forms the Srirangapatna island. This island has many religious and historic sites of significance.
There is even a train named after the Kaveri River, ( the Mysore-Chennai)  Kaveri Express, though the route of Mayiladuturai Express from Mysore to Mayiladuturai  better traces the course of Kaveri River.