I remember learning that you should say shema and aleynu together with the tzibur. If you're a little behind, do you stop what you're saying and say shema or do you catch up and say it when you're there? It would seem redundant to say it twice.
With regard to Shema, you should say at least the first verse with the tzibur even if you've already said it (Shulchan Aruch, Orach Chayim 65:2), and preferably all three paragraphs (ibid. 65:3 with Rama, see Mishnah Berurah 65:10). Similarly, if one did not yet recite Shema, one should recite it with the congregation, having in mind not to fulfill one's obligation by doing so (Mishnah Berurah 65:8), since it is better to fulfill the obligation when reciting the Shema together with its blessings. [This implies that you need to repeat the Shema when you reach the correct place.]
The reason is that if you don't say it with the tzibur, it may seem like you don't agree with what they're saying, ח"ו. Hence, even if you're unable to join them in the recital (e.g. if you're in the middle of reciting another blessing), you should try to use the same tune as them so it seems like you're joining them (S.A. O.C. 65:2).
The Mishnah Berurah (65:9) writes that you should also recite other prayers with the tzibur, such as Ashrei and Aleinu, presumably for the same reason (so you make it clear that you agree with what they're saying). However, it is unclear whether you need to repeat them when you reach the appropriate place.