To beat out the AI, I always make sure I've got the following in my production cities before I finish the Apollo Project:
Factories and power (usually Coal, since they're cheap and I can deal with a bit of unhealthiness)
Laboratories
Space Elevator
Usually, I'll have my #1 prod city building the Space Elevator (love having an Engineer to do this -- rushing to Fusion is nice for the free GE) while the #2 city does the Apollo Project. I keep the #3 city for the building of Mech Infantry and any extra tanks/MAs that I need.
Then I set the lower-output prod cities to build the easy space parts, like the casing. I don't want to waste the city that can build life support in 12 turns to build a casing part that will only get finished there a couple turns earlier than the lower-prod city. High prod cities do the big pieces.
Probably isn't a concern at low levels, but on Prince and above, the AI get's really pissy if they find out that you're near completion on the spaceship. Even a friendly civ might attack you to prevent your launch. So that's why it's nice to be able to finish it in BIG CHUNKS, rather than one piece at a time. Get your production set up so that all the casing pieces come together within a few turns. Also nice if you can first finish the expensive bits (life support, cockpit, docking bay), then quickly build the little pieces.