What is the proper way of quitting sunnah prayer when the Iqamah is pronounced; should one immediately quit the sunnah prayer and join the congregation or should one quit the sunnah prayer with tasleem just on the right side or on both sides, or is there no tasleem at all?

One has to cut off the prayer at the point that the Iqamah starts. However, if the Iqamah is made after a person has made ruku' for the second rak’ah of a nafil Salaah then there is no problem in completing that nafil Salaah because it will be almost over since less than a rak’ah  from it would be remaining.[1]

Making tasleem literally means abandoning the Salaah before it ends; thus you cannot make salaam because salaam comes at the end of Salaah to signify its completion.

And Allah knows best.

Source:

[1] [Shaykh Ibn Baz's fatwas: http://www.binbaz.org.sa/mat/1082]