What do I do with worn out Tzitzit from Tallit? Ganeza? Rubbish?
According to the principle halachah, there is no obligation to place the strings in geniza. However, lechatchilah the strings are not thrown away with the rubbish, out of respect for the mitzvah. If they were inadvertantly discarded, there is no obligation to remedy this.
Lechatchilah, the strings, and even the begged of the tzitzis, should be placed in the geniza. This is especially true of the tallis gadol, which is specifically used for tefillah.
It is permitted to use the tzitzis strings for another mitzvah, such as using them as a bookmark for a sefer. However, they should not be placed in a sefer for geniza purposes (which would be making forbidden use of a Torah sefer).
See Shulchan Aruch, Orach Chaim 21; Mor U-Ketziah 8 and Mishnah Berurah; Beis Baruch (on Chayei Adam) 11:258; Shut Shevet Ha-Kehasi Vol. 4, no. 10; Ginzei Ha-Kodesh Chap. 17, note 20.