Ash and Dawn shake hands in chapter two of 'The Ultimate Goal'. It happens when they are saying goodbye to Ambipom.
It is Ambipom that brings together the handshake by taking hold of both their hands with its tails to make them hold hands, because it knows that they are secretly in love. They shake hands for a while and then blush and let go.
'The ultimate goal' is a pearlshipping story of Ash and Dawn traveling through Unova and Kanto.
Ash is the main character in most Pokemon stories. His main goal in life is to become the worlds best Pokemon master, so he therefore leaves his home in Pallet Town in order to try and achieve this goal. On this adventure he meets lots of other people and Pokemon that try and either help or hinder him in reaching this goal.
Dawn is one of the people that he meets along the way. For a while she travels with Ash and Brock on her own quest to become a Pokemon Coordinator like her mother. And it is while they are travelling together that Ash and Dawn develop feelings for each other.
(Ash and Dawn with their first Pokemon while travelling together.)