KSRTC ( Karnataka State Road Transport Corporation), the state owned bus operator operates numerous bus services to and via Mysore.

For short distance travels you would not require an advance booking as tickets can be bought directly from the bus. KSRTC do operate a portal (www.ksrtc.in) that give lots of information on the bus services, timetable and even you can book seats online.

Let's first see the options to book a ticket in KSRTC. There are broadly two options , the first is from the KSRTC advance reservations counters (typically in the KSRTC bus stations) or through one of the many Franchisee Counters approved by KSRTC.
Advance Reservation Counters: Finding a Advance Reservation Counter directly operated by KSRTC is relatively simple as they typically operate from the KSRTC bus stations. Most of the major bus stations have one. There are about 70 such Advance Reservation Counters scattered all over Karnataka.
The  Franchisee Counters too operate more or less the similar way as above. The advantage is there are quite a few of them so that you will find one near to you. For example Bangalore city has some 150  KSRTC Franchisee Counters scattered all over the city center and beyond.
Other cities and towns in Karnataka to have  Franchisee Counters of KSRTC. You can even find  Franchisee Counters in the neighboring states too. Most of the outside state  Franchisee Counters are located in towns and cities to which KSRTC operate bus services.

Kerala has  KSRTC Franchisee Counters at Cannanore, Calicut, Kasargod, Ernakulam, Kottayam, Munnar, Palakkad, Payyanur, Tellicherry, Trivandrum, Trichur etc.
For Andhra Pradesh Franchisee Counters are available in Hyderabad, Secundrabad, Puttaparti, Srikalahasti, Tirupathi, Vijayavada and so on.

In Goa you can find a KSRTC Franchisee Counter at Panajim, Mapusa, Madgao, Ponda, Vasco etc.
Mumbai and Pune have many counters. Other Franchisee Counters in Maharastra is located in Aurangabad, Shirdi and Kolhapur to which KSRTC operate services

Tamilnadu has many in Chennai and Ciombatore. Also in places like Madurai, Pollachi, Ooty, Trichy etc. There is one available in Pondicherry too.
A full list and the locations, contact number of Franchisee Counters are available on KSRTC website.
In short in the southern states you'll easily find one advance reservation counter , either operated by KSRTC directly  or through one of the numerous approved franchisees.

KSRTC seat booking online : If you are located far or don't want to go to a reservation counter, the best option is to book it online at www.ksrtc.in

The online reservation works like this. You need to register your online account first. It's free and typical of such any online services. One that is done go to the reservation section (E-Ticketing) and login.
Go for Advance Ticket Booking and enter the source and destination places. The system will need the obvious details like passenger count, date of journey, class of bus etc. Once these details are entered search for the services. A list of available services are shown. Select one you prefer.
A trouble you may face here is about the selection of timing and the class. Because it will search only for buses of the class you opted and that too within the 'Journey Time Between' option you have selected. The 'Journey Time Between' essentially means the journey start time period. So it is better to know the available bus schedules, classes and start time so that it is much quick to book online.

Yes, they do have a search facility separately for locating services between a pair of destinations, map based search and also a timetable based on class. You can download the timetables as pdf file.
Back to the reservation process. Once a bus is selected from the list of possible options shown, you'll be presented with a seat layout where both booked and available seats are shown.
Select the ones you prefer and available. Along with many standard details, you need to select an "ID Card Type" that you are going to show as proof while traveling. That's probably to keep a check on bogus booking/ticket touts. There is a standard list of acceptable IDs (Driving Licence, Passport, etc ). Further is a typical payment screen using credit card or even the bank account.
Once booked, you need to print the ticket yourself and carry it while traveling along with the type of ID you've selected. The ticket with instructions are send to your email too.
Of course if you may want, you can later cancel the online booked ticket using the same login. KSRTC has a list terms and conditions for canceling. As a thumb rule, the early you cancel , the less money you lose.
KSRTC also have an online booking using your mobile phone (using ngpay). You will need to download some applications on your mobile. They have a demo for this on the website.
In short you can book seats in a KSRTC bus either online or by visiting one of the Advance Reservation Counters.