haphazard1
Dancing Bear
Research:
Research or Trade for the following Technologies in the order specified:
Civil Service -> Metal Casting -> Machinery -> Optics -> Astronomy -> Printing Press -> Scientific Method -> Physics -> Electricity -> Radio -> Mass Media
There will be no deviations from the beeline to Mass Media, except for Civil Service to run Bureaucracy and Liberalism to get free Scientific Method, Physics, Electricity, or Radio.
Trades outside of Beeline to Mass Media:
Guilds -> Banking to run Tokugawa's favorite Civic, Mercantilism
Gunpowder -> Chemistry -> Steel + Military Science to build Canons and Grenadiers
Gunpowder -> Replaceable Parts -> Rifling to build Riflemen
Nationalism to run Nationhood to draft Macemen, Musketmen or Riflemen.
I disagree somewhat with the beeline and others, especially since there is not any emphasis on which order to pursue them. A great deal depends on what (and when) we can get in trade.
Civil Service we definitely want, and it is not yet known to any AI so it will also make good trade bait. So this is our immediate target. But what to research after that? Metal Casting and Machinery are already known to two AIs -- researching these ourselves seems wasteful. I would prefer to research Liberalism after CS, up to 1 turn (or a fraction of 1 turn) remaining. No AI currently knows Education, or even Paper (I believe), so we have a window to research the tech and then see how far we can get before any AI is close and how expensive a tech we can take with the freebie. And the time spent on Liberalism will hopefully be enough that we can get Metal Casting and Machinery in trade.
We also need to keep our bulbing path in mind -- Gunpowder will open Chemistry to GS bulbing, which then opens Biology. Useful techs, and we may want to go that way, but it would be a significant diversion from our beeline path.
Additional non-beeline techs to consider/trade for: Drama (for Globe Theater in PigFish), Constitution (for our UB for happiness and EP)
Diplomacy:
Observe Worst Enemies and try to avoid trades with them when possible. Do not accept demands to stop trade, except possibly with Mehmed II or Isabella (we either don't need their Votes or in the case of Mehmed II we plan to make him our TUN Opponent.
As Sleepless has noted, we would like Mehmed as our opponent, but there is no guarantee this will happen. I think there is a decent chance -- Mehmed has a lot of land and more still available to expand into, so he should be large enough in population to be our eventual rival. A lot depends on the Justinian/Tokugawa war.
But let's not just write off Mehmed and (especially) Isabella. It will be difficult (or worse) to get either's votes, but not impossible. If we have to choose between them and someone else, we should definitely favor the other (more likely to vote for us) AI. But we should do what we can to improve relations -- resources, Open Borders, etc. We might get lucky with them asking for help that we can give, giving us a chance for some +diplo points.
Remain in Theocracy for Favorite Civic Diplomatic bonus with Zara Yaqob, Saladin, Isabella and Justinian I. Adoption of other Religious Civics should be avoided or kept as brief as possible (5 turns), since Favorite Civic Diplomatic bonus can decay rapidly.
Does anyone know the exact rule on this? I was under the impression that short periods spent in other civics would not significantly impact the shared Theocracy bonus, once we switched back to it. We may be able to use some brief periods in Free Religion, or Organized Religion.
Minimal effort (none) will be expended to improve relations with Mehmed II or Isabella.
As I said above, if we get a chance to improve relations with either that does not cost us with other (more likely to vote for us) civs then we should take it.
A trade to Zara Yaqob will be very lopsided to improve trade relations from 0 to +4.
A trade to Saladin will be slightly lopsided to improve trade relations from +3 to +4.
Getting +4 with Zara is likely to be difficult/expensive. The "free and fair trade" bonus is affected by the number of turns you have known a civ, and we have known Zara almost from the beginning of the game. If we can gift him something known to everyone (or almost everyone) else, great. But we should not give away anything too valuable, especially anything useful in war.
OasisWheat:
Grow to Max Happiness Cap 6P -> 12P
build Courthouse (whip) -> resume Longbowman -> resume Spearman (Engineering -> Pikeman)
Workers build two Grassland Farms (via Civil Service)
Keep an eye on hammer decay for the longbow and spear.
HorseCopper:
Grow to Max Happiness Cap 6P -> 9P
build Courthouse (whip) -> resume Longbowman
alternative University of Sankore (GS points if completed; 2 x Hammers input in Wealth if not)
Workers build four Grassland Farms (via Civil Service)
I do not think Sankore is useful for us -- we have no state religion, so it would be purely for the GPP. Better to run one more scientist rather than spending all those hammers. Sistine should be considered, as ianw1610 mentioned, just in case we end up needing to fall back on a culture victory.
Also, I do not know if the grassland farms are needed. Our happy cap is not so large that we could work all those tiles and the specialists the extra food would support. We will probably just work the river-side farms (FP and grass) and the grass copper. All other pop should be scientists.
PigFish:
Grow to Max Happiness Cap after whipping Courthouse
build - continue Settler -> Courthouse (whip) -> Harbor (whip)
Workers build six Grassland Farms
As with HorseCopper, I do not think we need the grassland farms. Work the food resources, the cows, and (maybe) the lakes. Everything else should be scientists, as our happy cap allows.
RiceMarble:
Grow to Max Happiness Cap 8P -> 9P
build - continue Courthouse (whip) -> Harbor (whip) -> Galley (whip)
Workers build seven Grassland Farms and one Floodplains Farm (via Civil Service)
Why another galley?
The floodplains (off the river) should be farmed, as it can support itself plus one specialist. But again the grass farms are probably not worth it at this stage.
Workers will preChop Forests before and after above mentioned Farms are completed.
The grass farms not on rivers are not an urgent priority, I think. They can be worked on, but over time -- there is not a huge rush to complete them, as we should be running Caste scientists from our best food tiles rather than marginal grasslands (taking 3 pop to get 1 scientist).
I'm still struggling to see how we can get mehmed to be our opponent? Justinian will win in his war against Tok unless we help Tok which loses us Just's votes and probably end up with Tok as our opponent instead of mehmed. Much better to reduce Mehmed to a couple of cities and win with Sal/Tok's/Zara's votes. With memhed backward we shouldn't have any problem rolling over him. My favoured war nowdays is Curr/Cav for speed and against LBs shouldn't be to hard.
I don't understand why you think Justinian will roll over Tokugawa? I forgot to check -- does Toku have iron? He has Machinery, so if he does he will have Protective crossbows which should give Justinian plenty of trouble. Toku also gets cheap barracks and free combat I for his melee units. He should not be an easy target.
As for wars...even better to win the vote with Justinian/Saladin/Tokugawa/Zara voting for us.

This leads on to another point used by Murky Waters in the previous SGOTM. Privateers? Could be a good use of getting GGs but obviously need chem/gp. Hopefully we will no more once our axe gets to the barb land mass.
Privateers would require Chemistry along with Astronomy, which would take us off our beeline by a good bit. They could be useful, so we should keep them in mind. But unless we get them before anyone else has Chemistry/Astro, they will just be sunk by frigates.
I think we should ensure that we have a Spy fortified in each of our cities, because we are at risk of our technologies being stolen and then traded around.
If we want to consider the alternate victory condition (Cultural) we need to make a decision now, as the Sistine Chapel is still available to build...
Stealing techs takes a lot of EP, well more than any AI should be able to manage at this stage of the game. Unless someone gets a Great Spy, of course -- we should watch the turn updates carefully in case anyone generates one, and check our EP ratios. If we get infiltrated, it will be obvious in the EP ratios.
Some spies could still be useful to guard our cities against poisoned water, etc. They cost hammers and unit maintenance, though. Short term I would like some more military, especially some longbows, to boost our power rating and give us something resembling real defense.

Building Sistine somewhere just as a backup plan is a good idea. We will be running lots of specialists, and the extra culture could be useful for tile control and defense.