Question: I live within an eruv. Can I carry my tallis home after Shul even though I won’t be using it before next Shabbos?
Answer: The Mishna (Shabbos 113a) teaches that one mustn’t prepare one’s bed on Shabbos for after Shabbos as it is a prohibition of hachana, preparing. Thus, the Shulchan Aruch (OC 323:6) allows one to wash dishes that they will use again on Shabbos, but not for use after Shabbos.
Accordingly, one would not be able to take one’s tallis home on Shabbos because he wants to take it to a different shul the following day.
The Mishna Berura (290:4) writes that one shouldn’t say that they’re sleeping on Shabbos in order that they will be able to work after Shabbos. This applies to all forms of hachana if it is clear that one is doing so in order to prepare for after Shabbos. R’ Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 28:77) writes that this even applies to mitzvos. Thus, one may not prepare for havdala while it is still Shabbos.
R’ Shlomo Zalman Auerbach (quoted in Shemiras Shabbos Kehilchasa 28:89) and R’ Ovadia Yosef (Yalkut Yosef, Hachana Mishabbos Lechol 12) maintained that things that one regularly does as a matter of course may be done on Shabbos even if it may be preparing for another day. Otherwise, one wouldn’t be able to put books back on the shelf after using them! Thus, one may take one’s tallis home after wearing it. Likewise, one may carry one’s keys even though they won’t be used until after Shabbos.
In conclusion, one may take one’s tallis home after wearing it providing that they are not specifically doing so in order to use it after Shabbos.