hieromonk---HIGH-aero-munk said very fast, kinda like how you say Cairo, Egypt..
orthros----OAR-thros (long O)
compline----CAHM-plin (long I)
matins----MA-tins (short I)
oekonomeia----e-cah-nah-MEE-ah (like economy with an "ah" at the end)
theologumenon----THE-o-LO-goo-men-on (TH like wiTH, not THat.)
St. Gregory Palamas----PAHL-ah-mahs
St. Basil the Great (BAYsil or Basil?) Just like the normal way--Americans say Bay-sil, British say Basil.
Where do you live? Have you gotten in touch with a priest in your area? We can help you find parishes if you wish.