I went through 20 turns which I believe is the standard for a start.
4000 BC- I can see a flood plain wheat through the fog just south of the starting location. Move the worker and settler South.
3950 - Found Rome with 2 flood plain wheats in the city limits. Set research to writing at minimum science rather than the usual pottery since we have so much more available food than shields. Worker moves to the flood plain wheat and begins irrigation.
3700 - Rome grows, luxury tax to 10%. Worker begins roading the wheat.
3600 - Rome completes the first warrior which heads east. We are already hit by disease

!
3550 - Luxury dropped to 0 as we are size 1 again. Worker heads to mine a hill during despotism! The only shield producing tiles we have are hills or desert, so we need more workers to make the land productive or resort to the whip.
3500 - Spot a hut, but avoid popping it because we can't afford barbs right now.
3450 - Growth back to size 2 and luxury tax raised.
3300 - Rome completes another warrior who heads West.
3250 - The western warrior spots coast and an ivory resource. Rome is hit by disease again canceling a 2->3 population growth.
3200 - Pop a hut further east from Rome and get barbarians. The warrior (on a hill) survives and promotes to elite. Rome is hit by disease and is size 1 again

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3150 - Lux turned off as it isn't needed any more.
3050 - Contact with Egypt. Trade Egypt Alphabet and Warrior Code for Ceremonial Burial, Bronze Working, and 10g. (Spears will be necessary soon).
3000 - Rome grows to size 2, luxury tax to 10%.
Um, I guess this would qualify as a slow start

. The save may be obtained from the following link:
http://www.civfanatics.net/uploads4/EM-5_3000bc.sav.zip