Most definitely.
Flying on Indian domestic airlines, is as safe as flying anywhere else and certainly safer than travel on the Indian highways.
That said, while most popular tourist destinations have airports in close proximity and flights between destinations are pretty frequent, it is not guaranteed that all destinations on your travel plan will all have airports. Hence, you may have to plan your transportation to include part air and part road/rail modes of transport.
This website gives pretty good pointers on the best route and transport options between cities.
Some of the popular domestic airlines are Kingfisher Airlines, SpiceJet, Jet Airways, GoAir Airlines, IndiGo. You can compare the fares/flight details and book for flights on websites like Makemytrip , Yatra, Cleartrip .
Quite often airlines introduce new domestic destinations on their travel routes, at really low introductory prices, it might be a good idea to check for such offers before you book your transport.
Based on popular opinion and traveler reviews, it is suggested that you prefer privately run airlines over the government run Air India Limited flights ( Air India and Indian), as these are notorious for poor service and quality of flight, despite the government putting in vast efforts and resources to change things for the better.