I am planning to buy the tickets online on my own for domestic flights from Bangkok to Phuket. Can you please recommend a domestic airline? (Please note it has to be reliable, as I will be connecting to international flight on the way back).

For me the most reliable airline would be Thai Air. They are a domestic and international carrier. They are a full service airline.
Thai Airways

Sample schedule:
ROYAL e-booking - air info

There are some discount airlines that also fly to Phuket. If you go with a discounter make sure you know about their weight allowances. There are some restrictions on weight and size that a full service carrier doesn't have.

Allow enough time between flights for connections. My advice is to stay a few days in Bangkok at the start of your trip if you can. Why fly all the way to Thailand and not see the capital city? You can even purchase flights to Phuket from local travel agencies if you want. Even Bangkok has warm weather! Once you get away from the tall buildings and can find some sunshine you can even get a tan!

Some tourists prefer to spend the last night in Bangkok where they can take in a dinner cruise, see the city and do some last minute souvenir shopping, etc. This way they are only a short taxi ride to the airport. From the city to the airport figure on 35 - 55 minutes. Taxis in Bangkok are cheaper than Phuket and available 24 hours a day.

Good luck.

I booked a flight recently Phuket to Bangkok with Nok Air for 1,500 Baht. Pretty cheap!

Other airlines you might check out:

one-two-go (ÊÒ¡ÒúԹâÍàÃÕ¹·ìä·Â : ÊÒ¡ÒúԹ¨ÃÔ§ã¨... à¾×èͤ¹ä·Â.)
bangkok airways

Does Air Asia fly to Phuket? Not sure but may be worth checking. As above though check the small print on any ticket you buy. You may end up paying more than you wanted to because of some charge or another. Good luck and have fun!

If you need to connect with an international flight check with airline to see who they have arrangements with to check baggage all the way through , this will give some indication who to book the domestic flight with .
I would book with Thai for their reliability and luggage allowance , may cost a little more but a lot less than having to pay for another ticket if you miss flight .

A few good domestic airlines exist in Thailand, Nok Air, Bangkok airways , Airaasia are the 3 main ones.