Here is a list of popular excursion points from Mysore. Checkout the routes options, distances from Mysore city and build your itinerary.
Srirangapatna.

The island of Srirangapatna is located some 15km northwest of Mysore on the Bangalore to Mysore highway.

There are many attractions in Srirangapatna including the Ranganathaswamy temple, after which the island is named, Srirangapatna Fort built by Tippu Sultan, Daria Daulat the summer residence of Tippu Sultan, Gumbaz , the mausoleum of Tippu and a large number of historic sites scattered all round the island. Apart from the Ranganathaswamy temple, Srirangapatna has many other places of religious significance. Karighatta hill top temple and the Nimishambha temple are popular among the pilgrims. So is Paschima Vahini, the confluence of Cauvery river at the end of the island.

Brindavan Gardens
Brindavan gardens on at the KRS Dam is an immensely popular tourism spot. It's about 20km northwest of Mysore city. You can travel to KRS from Mysore city as well as bypassing the city.

Bylakuppe The Tibetan settlement at Bylakuppe is quit a fascinating location, an opportunity to experience the Tibetan culture in this part of the country. This is an important learning center of Tibetan Buddhism outside Tibet. The Golden Temple is spectacular along with other schools of Tibetan Buddhism. Bylakuppe is about 2 hours drive (80km) from Mysore towards Madikei (Coorg region).

Somnathpur (east of Mysore) :

Towards the eastern side is the ancient capital of Talakadu with numerous temples buried under sand dunes.

Near to Talakkadu is Somnathpur, known for the Hoysala style Keshava temple. If you start early, Sonmathpur and other attractions in the city can be covered in a day. It will be quite a whirlwind and needs your private transport.

Further east of Somnathpur is Shivanasamudra (Bluff) with the twin waterfalls - Gaganachukki and Barachukki ( Bharachukki). Mysore to Somnathpur distance is about 40km. Somnathpur to Talakadu is some 30km. Bluff is another 35km northeast of Talakadu.

Somnathpur-Shivanasamudra-Talakadu make a triangular route ideal for a days trip, if one can start a bit early from Mysore and prepare for a bit hectic schedule. Both Shivanasamudra and Talakadu involves some trek. The road conditions near Somnathpur is not that great, add an extra 30min.
Also if you are traveling back to Bangalore direction, consider the NH209. You'll save time in driving back to Mysore.

Bandipur