Catch that overnight Cauvery Express from Chennai (Train No: 16222 & for return Train No: 16221). This is a great options in terms of cost and time to travel from Chennai to Mysore.
If you are looking for a day travel aim at the Shatabdi Express. Though relatively expensive Shatabdi Express takes you from Chennai to Mysore in 7 hours flat, rivaling any other options. Shatabdi Express leaves Chennai (MAS) early in the morning (6 am) and reaches Mysore (MYS) by 1 pm. Also keep in mind on Wednesdays Shatabdi Express does not operate on either directions.
There are daily bus connections between Chennai and Mysore. Of which 5 are Volvo (AC) and 3 are Rajahamsa, both operated by KSRTC. The Tamilnadu State Express Transport Corporation too operate trips to Mysore from Chennai.
If you break your Chennai to Mysore journey into two laps, keeping Bangalore as the transit point, the options are a lot more than the above two trains or the direct bus services. There are many trains (even buses) between Channai and Bangalore. And Bangalore to Mysore is a well connected route by both train and bus.
In short if you are not able to get book a seat in the Cauvery Express, Shatabdi or in one of those direct buses, travel to Bangalore from Chennai. Then do the onward journey from Bangalore to Mysore , either by road or rail.
Chennai to Mysore is well connected by a series of highways (about 500km). See Chennai to Mysore by road.
Book well in advance as the train and bus connections between Chennai and Mysore usually get over booked, especially during the weekends and also during those holiday seasons.