As Jo Grant explained to Latep the Thal, a handshake was an "old Earth custom" which entailed the clasping of hands between two people, showing that they were good friends and that we were pleased to meet one another. (TV: Planet of the Daleks)
The Fourth Doctor begrudgingly shared a handshake with the Tremas Master as the two Renegade Time Lords agreed to work together in order to save the universe on Logopolis. (TV: Logopolis)
Tegan Jovanka and Vislor Turlough shook hands as the latter chose to leave the TARDIS. (TV: Resurrection of the Daleks)
Following the destruction of Davros' right and last hand, the Sixth Doctor facetiously offered him a handshake. (TV: Revelation of the Daleks)
Martha Jones and Donna Noble shook hands as they were introduced to one another by the Tenth Doctor. (TV: The Sontaran Stratagem)