10 turns (75 BC - 175 AD)
I'll just start from the beginning with Great People. Sorry if any of it is too basic, but better too much explanation than too little. A Great Person is born in a city whenever the Great Person meter maxes out. So how can we do that? Three ways really. First, certain wonders give you a certain number of Great Person points per turn. Additionally, these points are always of a certain type. For example, The Hanging Gardens give you +2 Great Engineer points per turn, while The Collosus gives you +2 Great Merchant points per turn. Your first Great Person requires 100 total Great Person points in any one city. When you finally accumulate 100, a Great Person is born. The percent chance of creating a certain type of Great Person is dependent on the type of points (engineer, prophet, artist, scientist, merchant) generated. After the first Great Person is born, the requirement to make a new one is increased to 200 total points. And each time after that, the requirement goes even higher. Here's where our civ is right now.

We currently have 120 out of 200 Great Person points required to generate the next Great Person. The requirement is 200 because we already made a Great Prophet when the slider hit 100. We are generating 8 Great Person points per turn in Mecca. How? The Oracle, which we built in Mecca generates +2 prophet GP points. The Hanging Gardens generates +2 engineer GP points. Our base GP point generation per turn is +4, which doubles for us because we are using a philisophical civ. So why does it say that we have a 73% chance of making a Great Prophet and only a 26% chance of making a Great Engineer at this point? Because The Hanging Gardens were just built recently and The Oracle has been making us prophet GP points only for a little while.
Ok, here's the long term plan for the Great Person farm in Medina. Great Person farms are based on working specialists in a city. What are specialists? They're these guys.

Note that I'm not actually working any specialists here as all the icons on the right are blank. I just posted the picture to show what the specialist does. An artist specialist generates 1 beaker, 4 culture, and 3 Great Person points per turn. 3 Great Person points per turn are a lot. That's even more than a single wonder grants you. So the gist of the Great Person farm is to work as many of these specialists as humanly possible. The problem with specialists is that they use up food while not actually generating any. Whenever you employ, say, an artist specialist in a city, it's like working a foodless tile that says "+1 beaker, +4 culture, and +3 Great Person points". It's almost like a mine or a quarry for culture and Great Person points instead of hammers. Now as you've seen by now, you can't have a city full of just mines because you aren't working any food tiles to sustain your population. The same goes for specialists. To use a lot of specialists you have to have a lot of food. That's why we want to be building farms everywhere we can in Medina. But food isn't the only cap on population. If your city becomes too unhealthy or too unhappy, your people will sit on their asses instead of working a tile or working as a specialist. This is why we want to focus on health buildings (granaries, aquaducts, The Hanging Gardens) and connect health resources. So then, what are we going to do about happiness then? This.
The Globe Theatre
+6 culture, +1 artist GPP
No unhappiness in city, can turn 3 citizens into artist.
Requires: Drama, 6 theatres
Once this is built in Medina, we will never have to worry about our people getting unhappy there again. We'll just have lots of artist specialists and farms in order to feed them. Additionally, you should know that you can't just hire specialists wherever you want. You have to meet certain requirements. As you can see The Globe Theatre allows you to employ 3 artists. A building like a normal theatre lets you employ 2 more. So the types of buldings you have in your city are another cap on how many specialists you can run in a city (unless you use the caste system civic, which removes the cap on the number of artists, scientists, or merchants you employ). So just to sum up, if we want to make a Great Person farm, we want lots of food, health, happiness, and things that allow us to use more specialists. Oh, and more mltipliers (besides philosophical trait, which we already have).
National Epic
+4 culture, +1 artist GPP
+100% Great Person birth rate in city (adds on top of philosophical bonus)
Requires Literature, library in city
The Parthenon (already built elsewhere, oh well)
+10 culture, +2 artist GPP
+50% Great Person birth rate in all cities
Requires Polytheism
Pacifism civic
+100% Great Person birth rate in cities with state religion, +1 gold support cost per military unit
Requires Philosophy
We're going to want all of these eventually (except The Parthenon, which was already built of course).
Alright, with that out of the way here's what I did during my turn. Because Medina is going to be the Great Person farm, I started building an aqueduct there instead of the library. Also started on a work boat instead of a library in Mycenian. Right now, libraries don't do much in either of these cities. Libraries give you a +25% bonus to science in the city. Medina is currently making 3 beakers a turn and 25% of that is less than 1. Mycenian is at 1 beaker per turn so building a library there right now does nothing. Eventually libraries might be decent in these cities, especially Medina since it builds culture, but I think right now we should focus on growth. Also, I think we had our workers working in reverse. One was building a farm near Mecca, where I think we'd be better off building cottages. The other worker was building a cottage in Medina, where pretty much all we want are farms whereever they can be built. Great Person factories are all about food and health. I decided to send both workers south to Damascus in order to improve tiles and speed up production of the Sistine Chapel (+2 culture per specialist per turn! That's A LOT more culture in the Great Person farm).
So here's my take on where we're at after my turns... New settler is headed to that space to the west, one tile north of the cow, and one tile southwest of the deer. After the city is founded, we should start on a theatre there. Theatre's are better value than obelisks, as they're still pretty cheap to build and provide +3 culture rather than +1. We should also chop down that jungle in Damascus and build a mine on that hill. I'll leave the rest up to you. Things we could use right now are more military, more theatres (we need 6 total before we can build the Globe Theatre), and it would be nice to finish the Sistine Chapel just in case the computer gets the funny idea to finish it using a Great Engineer. Oh, and I just wanted to point this out.

I'm pretty sure both of the white cities shown will be ours sometime in the future. They will succumb to our cultural might, just you wait. (If these cities do happen to flip to us during one of your turns, destroy them, as they are in horrible places.)