><> and CJam, 165 bytes
"~~~~~~~~~~~~~~~~~~~~~~~r00gol?!v93*0.Hi
' < .1*5av!?log10oar~~~r
'"`{"`"\"_~e#.21 <.2+4*96;!?log10oa"}_~e#.21 <.2+4*96;!?log10oa
To CJam, the program starts with a multi-line string literal. This is escaped with `, and then it uses the standard quine to print the quine code, as well as a trailing comment.
To ><>, the first " starts a string literal that goes through the entire first row, pushing every character to the stack. After that, the trailing spaces (created due to the input being padded) are deleted, and then the stack is reversed. Every character in the stack (i.e. the entire first row) is output, and then it moves down to the second row.
The second row essentially does the same thing, except that it's in the opposite direction, so you don't need to reverse the stack. (I do anyway, because I have to delete the trailing spaces.)
Finally, it moves on to the third line. The only major difference is that you must skip the CJam block, which is done using . The single quote captures the entire line (again, backwards), and then it is output.