Showering and Shaving Before Prayers
The Rabbis only prohibited one from getting a haircut or going to a bath house a half hour before the time of Mincha arrives, since this was common practice. It is therefore permitted to get a haircut and to enter a bath house before dawn, which is when the time of Shacharit actually begins (Shulchan Aruuch Orach Chaim 89:7), but after dawn one should not do so before praying. (Mishneh Berurah in the name of Eliya Rabba)
The Piskei Teshuvot explains that the above prohibition only applies to bathing that is similar to a bath house, which would take a while, and poses a concern that one might take too long or become too tired to pray. A shower (or bath — Halacha Berurah), which is common in these times, is permitted, and considered like washing one’s hands, face and feet in honor of
The Piskei Teshuvot, citing Halacha Berurah, permits one who is accustomed to shaving in the morning to do so before prayers, even after dawn and sunrise. He explains that today there is no concern of the utensil breaking (which would require fixing it), and it also does not take up a lot of time. He adds that one doing so should first recite the blessings on the Torah and say Kriat Shma. In the footnotes he cites two opinions which forbid shaving before praying: Halichot Shlomo in the name of Rav Shlomo Zalman Auerbach, and Rav Ben Tzion Aba Shaul in his Sheilot VeTeshuvot, and he therefore concludes that one should be strict unless it is really necessary.
The Mishneh Berurah writes that there are those who prohibit doing other types of things that are generally done in the early morning, and therefore it is advisable to say morning blessings before doing so.
