What to Do When Arriving Late - Part 1
If one arrives at the synagogue and finds the congregation approaching the end of the “Verses of Praise” (Pesukei d’Zimra), he should say “Baruch She'amar”, followed by “Tehillah LeDavid” (Ashrei). Next he should say “Hallelu et Hashem Min HaShamayim” and then “Hallelu E-l B’Kodsho, and afterwards he concludes with “Yishtabach” (for what to add if there is more time, see Rema's comments to 52:1. If there is not enough time to say all of the above, one should also skip “Hallelu et Hashem Min HaShamayim”. Rema: If there still is not enough time, one may say only “Baruch She’amar”, “Tehillah LeDavid” and “Yishtabach”. At this point one should continues with the Shema and its blessings, followed by the Shemoneh Esrei together with the congregation. (Shulchan Aruch Orach Chaim 52:1)
It is of the utmost importance to try to arrive early to prayers in order to be able to say all of the morning prayers in their proper order. In fact, “The Maggid” (an angel that taught Rabbi Yosef Karo (author of the Beit Yosef and Shulshan Aruch) warned Rabbi Karo about this, explaining that one who leaves out parts of the prayers mixes up the “spiritual channeling” from Above. This is a very serious matter, understood by those who know the secrets of the Kabbalah (Be'er Hetev citing Siddur Arizal; cited by the Mishneh Berurah, Kaf HaChaim and others).
Based on the above, many great rabbis and men of outstanding accomplishment in avodat Hashem (Divine service) would not skip any parts of their prayers even if they came late at times. According to them it was more important that their prayers be spiritually intact than be together with the congregation when faced with this less than ideal choice (Be'er Hetev; Petach HaDvir). The Kaf HaChaim writes that several rabbis rule this way and that he agrees with them, stating that there are many deep and wondrous secrets contained within the prayers, and one should therefore not skip any one of them.
The Mishneh Berurah rules like the Chacham Tzvi who disagrees with the above ruling. He maintains that even according to the Zohar one who prays with the congregation should skip in order to pray the Shemoneh Esrei together with them. Yalkut Yosef explains that the ruling of the Be'er Hetev and Kaf HaChaim applies only to those who always follow the Zohar and Arizal (this includes many Sefardim and Chassidim); however the halacha is in accordance with the Shulchan Aruch as stated above. According to this approach the great advantage of praying together as a congregation overrides everything else.