It is a rabbinic obligation to recite Keriat Shema of Shacharit with tefillin, as written in the paragraph of Shema, “And you shall bind them as a sign on your arm and they shall be as frontlets between your eyes” (Deuteronomy 6:8). Similarly, it is written in the paragraph of V’hayah Im Shamo’a, “Bind them as a sign on your arm and let them be as frontlets between your eyes” (Deuteronomy 11:18). Thus, it is not proper to read these verses without tefillin. The Chachamim teach, “Whoever recites Keriat Shema without tefillin, it is as if he testifies falsely about himself” (Berachot 14b).
In any case, even one who does not have tefillin must recite Keriat Shema, since tefillin and Shema are two separate mitzvot that do not prevent one another from being fulfilled. If he does not have the privilege to perform the mitzvah of tefillin, he must at least fulfill the mitzvah of Keriat Shema, and he is not considered one who gives false testimony, since he is in circumstances beyond his control (Mishnah Berurah 46:33).
It is appropriate to pray the Amidah of Shacharit with tefillin as well; that is part of the complete acceptance of the yoke of Heaven (Berachot 14b).
Regarding a person without tefillin, whose friend can give him his pair to put on after he finishes praying, the Acharonim are uncertain what is the best course to follow. Is it preferable for him to pray in a minyan without tefillin and then put on tefillin afterwards? Or should he pray with tefillin individually after the minyan? The opinion of most poskim is that in practice it is best to pray with tefillin individually. However, one who wants to pray in a minyan without tefillin and then put on his tefillin afterwards is permitted to do so.
Likewise, it is proper to put on one’s tzitzit before prayer because the third paragraph of Keriat Shema discusses the mitzvah of tzitzit and it is appropriate to recite it while actually fulfilling the mitzvah of tzitzit. It is also customary to hold the tzitziyot and kiss them a number of times while reciting Keriat Shema (see further in this book 15:11). Even though the mitzvah of tzitzit can be fulfilled by wearing a “tallit katan” (tzitzit), it is also customary to put on a “tallit gadol” (tallit) in honor of the Shacharit prayer. However, before marriage, most Ashkenazim suffice with wearing a tallit katan. Only after they marry do they begin to wear a tallit gadol for Shacharit.