Thanks for the advice, Izzy. I sprayed all the blooms with water and got a lot of the bugs off that way. Then I went over each one with a Q-tip to dry it and check for more bugs and remove them by hand. I sprayed the stems and the leaves with my soap/water/cinnamon solution.
Should I still spray the whole plant with neem oil or oil/soap solution, blooms and everything? Also, where do thrips come from? Are they somewhere in my apartment, or were they most likely on the orchid when I brought it home?
Thanks again for the advice.