The Shulhan Aruch (O.H. 89:3) rules that one may not engage in one’s affairs prior to praying in the morning. The Shulhan Aruch is not referring to those activities in which one must engage in order to prepare for prayer, such as washing up, driving to the synagogue and the like. Rabbi Ben Zion Abba Shaul (Or Lezion, vol. II, ch. 45) says that stopping by a store on the way to the synagogue in order to buy a needed item is permissible and may even be considered a Mitzvah.
Rabbi Shlomo Zalman Auerbach (quoted in Tefila Kehilchata, ch. 6) says that one may engage in simple Melachot (labors) such as placing one’s clothing into the washing machine.
If one must engage in some sort of work or personal affair, one should at the very least recite all of the Birkot HaShahar.
Similarly, after Birkot Hashahar one can quickly answer a work-related email or text as long as it mostly effortless and does not lead to an engaging exchange (c.f Eshel Avraham 89). More engaging labors such as actively doing the laundry or sitting down to do work-related activities is not permitted.
Summary: One may not work or engage in one’s affairs prior to praying in the morning. One may engage in simple tasks, as needed, prior to praying.