I am vaguely aware of the Pigeonhole principle and I understand that you would need 367 people to ensure that two people have the same birthday. I think that it may be required to have 734 people in a room to ensure 100% probability, since you can have 366 birthdays repeated and not have three people. Is this a correct assumption, and if so, how would you solve it without just guessing/checking?
Your reasoning looks pretty much right (you're off by one) and looks to me like a direct approach more than guessing and checking. Maybe if we just crystalize your argument a little, you'll understand it more deeply:
There are $366$ possible birthdays; if we have $366\times 2=732$ people, then it is possible for each day, exactly two people call it their birthday. This is the maximum number of people where we can have at most $2$ per birthday. If we have one more person, then at least three people must share a birthday. Thus, in a room of $733$ people, at least three share a birthday.