Really? I never do that. I'm scared all decent tenets will be taken by the time I can earn enough faith to get a missionary and a second prophet.(typically before enhancing).
I'll sometimes build a missionary first if I'm counting on Hagia Sophia or finishing piety to trigger my last prophet. If so you're wasting your time waiting, better to start spreading as soon as possible to get the bonuses at least to the rest of your own cities. Lately I've been getting enough faith this is too long to wait though, and in this case I save it all for that 2nd prophet to get the most options on beliefs. I never build the buildings before either enhancing or missionaries though UNLESS I get borobadur because that gives me all the missionaries I need in the short-term.
This brings up another question: If you're using the Hagia Sophia or Piety closer for the 2nd enhancement, is it better to start building Missionaries or wait for the 2nd (300 faith) Great Prophet for a 4-conversion unit that doesn't suffer Attrition, rather than the 2-conversion, Affliction-susceptible Missionary?
This is only partially true. I find the game sort of missed the balance of this, because:Actually using missionaries first to spread your religion to civ which don't have own religion is quite good. They defend your religion more than you do. In my current Venice campaign, Russia proposed my religion to be world religion in World Congress.