There are 2 ways to get to Koh Tao from Bangkok. There is the slow, cheap way (by bus and ferry) which will take more than 14 hours or the quicker, not so expensive way to get there (fly and ferry).
By Bus and Ferry
There are plenty of buses that travel from Bangkok to Suratthani and then you can transfer to a ferry across to Koh Samui, then you can change ferries go from Koh Samui to Koh Tao via Koh Phangan. The bus trip can take 10 or more hours. The ferry trip from Koh Samui to Koh Tao take around 2 hours or more depending on sea conditions.
You can also get a bus from Bangkok to Chumphong which has ferries to go direct to Koh Tao, though these ferries are not as regular as the ferries from Koh Samui.
The bus trip from Bangkok to Suratthani will cost just over 500 Baht. If you pay 700 - 800 baht you can get the bus trip and a connecting fery to Koh Samui included.
By Plane and Ferry
The quickest way to go from Bangkok to Koh Tao is to fly from Bangkok to Koh Samui which takes around 1.5 hours to fly. You can then jump on a ferry to Koh Tao via Koh Phangan.
Traveling from Bangkok to Koh Tao