Since Beyond the Sword has an expanded Modern Age and switched around all of the techs for the various spaceship parts, it makes sense to completely re-visit the entire thing in a new post. So here is my tech path for BtS 3.0.3.- (all spaceship component costs are listed for Epic speed)
0) Rocketry - This of course unlocks the Apollo Program. Really, any city can build this now. The entire path takes so long to research that it doesn't really matter how quickly you build this. Heck, you can't build any spaceship parts yet anyway.
1) Plastics (needs Industrialism) - Plastics gets the bump because it's required for Computers now. This comes early enough now that it might be worth it to build the Three Gorges Dam. Normally I avoid this because my major production centers already have Coal Plants, but we still haven't unlocked any spaceship parts yet. If you're dying to build something, then have fun.
2) Computers (needs Radio) - The Internet gets moved to this much earlier tech. It's still too expensive for a minimal gain, but like I said before, we haven't unlocked anything yet. I don't know the AI's preferred modern tech path yet (my only BtS space win was a runaway), but this might gain us a tech or 2 if they go a different path. I still say it's not worth it.
3) Superconductors (needs Refrigeration) - This tech is important because it unlocks the Labortaory. Every city needs this improvement because of its 25% research bonus as well as its 50% bonus to spaceship part completion. We also unlock our first piece, the Thrusters (1800
, 2x speed with Aluminum). We need 5 Thrusters, but build these parts in lesser production centers. Even if they take 30 turns to build, we still have plenty of time. Get those Labs done first, though!
4) Satellites - Here we unlock the Docking Bay (3000
, 2x speed with Aluminum) on our way to Robotics. The Docking Bay is actually the most expensive piece to build, but the Aluminum bonus makes it a little easier than the Engines. Again, we still have 6-8 more techs left, so this doesn't have to be done by one of your best cities. Keep building the Dam and the Internet and let a lesser city build this piece.
5) Robotics - Here is the Space Elevator. Personally, if you're in a tight space race, this tech can be skipped altogether. It's a dead end and the Space Elevator is vastly overrated. An extra 50% is not worth the hammers. It does let you build Mech Infantry, so if you're worried about defense you need this tech.
6) Composites - This unlocks the Casings (1800
, 2x speed with Aluminum) as well as everyone's favorite Modern Armor. Like the Thrusters, these should be built in minor production cities. You have the option of building 1-5 Casings, but anything less than 5 gives you a chance to fail the launch. It's completely ridiculous to not build 5 Casings. Even if they take 30 turns in your minor cities, you still have lots of time.
7) Laser - A new tech that doesn't unlock anything but is on the path to Fusion.
8) Fiber Optics - Here's the Cockpit (1500
, 2x speed with Copper). It's a pretty easy build, so again let a lesser city build it. Your top 3 production centers are needed for the last few pieces.
9) Fusion (needs Fission) - You get a Great Engineer for this, so use him to finish up the Three Gorges Dam or the Space Elevator. He can't help you with the spaceship parts. This unlocks the Engine (2400
), the hardest piece to build. You only have to build 1 Engine, but you have the option of building 2. Building 2 Engines depends on how good your production centers are. You need your 2 best production centers to work on the Engines. In my test game, the 2nd Engine knocked the arrival time from 19 turns to 16 turns (Epic speed, remember). It's highly unlikely that you can build the 2 Engines and the Stasis Chamber (with the next tech) quickly enough to make the 2nd one worthwhile.
10) Genetics (needs Medicine) - Here's the next to last tech and the next to last piece, the Stasis Chamber (1800
). Plan this build so it finishes roughly the same time as the Engine. If your 2nd best production center is much worse than your best city, it might be better to have it build the Engine instead. You still have 1 piece left, so have a city ready for...
11) Ecology - Here's the Life Support (1500
, 2x speed with Copper), a fairly easy piece to build. If you time it right, the Engine, Stasis Chamber and Life Support should all finish around the same time. If you can do that and still squeeze in a 2nd Engine, then more power to you. Now all you have to do is survive until the ship reaches Alpha Centauri!
Other people have listed ways to speed up production if you have a piece lagging behind the rest. After Ecology, Towns are useless, so you can plow them under for Workshops. Although you can't rush buy or slave pop for spaceship parts, you can still chop forests for them (wooden Engines??
). Also, the new espionage system allows more ways to stop other civs than ever before. Just remember, for stopping other civs, you only have to stop 1 essential piece to stop the whole thing. Usually that's the Engine, but it can also be the Stasis Chamber.