Koh Tao is a small island so one can reach by taking a flight to Koh Samui or Chumphon and then a ferry.
Major cities in India have flight connection with Koh Samui, although that is an expensive option. One can take a flight from India to Bangkok and then take a bus to Surat Thani/Chumphon, and then take a ferry to Koh Tao for a cheaper alternative. Mumbai, Delhi, Chennai, Ahmedabad are a few of the major airports that have flights to Bangkok.
Tourists have the option of flying to either Koh Samaui international airport which is the simplest but very expensive and not usually a choice for budget travellers, or Chumphon international airport which is much cheaper than Samui. Both these airports have high speed ferry connections to Koh Tao.
If you're coming from Bangkok to Koh Tao, you can take a bus from Bangkok to Chumphon and then a ferry to Koh Tao. This is a very long journey and hence not usually prefered. But it might appeal to people travelling on a budget.
Boats are the only way to reach Koh Tao. Services run from the mainlands of Chumphon and Surat Thani, and from islands of Koh Samui and Ko Pha Ngnan.
Walking around on foot in Koh Tao is the best way to explore the island. Taxis are available too, but they are a rather expensive option. Another alternative is motorbike and cycle rentals. The mountainous geography of the island makes travelling across it rather time-consuming and arduous, so make sure you account for that in your plans.