When should I say Birkot haShachar? Should I say the brachot that appear at the beginning of the siddur consecutively in shul at the beginning of davening or at home?
Birkoth haShahar should be said as one performs the relevant action. For example, after one opens one's eyes upon awakening one says ברוך פוקח עורים. When one gets dressed one says מלביש ערומים. See Talmud Bavli Brachot 60b, Rambam Tefillah 7:7, Rosh Brachot 9:23, Tur and SA OH 46.
Hazal provide us with an instruction manual for getting up and going in the morning. If we follow the directions in that manual as intended, getting up, getting dressed, etc., and reciting the brachot will be much more meaningful. The impact these brachot have on a person when each is recited in its own time is very much greater than when said afterwards; one appreciates Hashem's blessings so much more. Try it and experience the difference!
Nevertheless, if one did not say the brachot at the appropriate time, one must still recite them later. See Siddur Rav Saadia Gaon p.89.
17th of Adar II, 5771 Wednesday, 23 March 2011