EARLY ERA EGYPTIANS

1. Opening moves
Invade Syria on the first turn with everything you can, leaving garrisons in only Antioch and Egypt.
After that, many players wipe out the Turks immediately. I garrison Antioch and Syria and wait a turn or two. There is a good chance the Turks will offer a ceasefire and then get into a war with the Byzantines. Then you can attack again and wipe them out with less trouble.

2. Units to train
Desert archers will help you defeat Turkish horse archers, and are a good unit throughout the early era.
Bedouin camels are not fast enough to make an ideal counter to horse archers. However, they will destroy bodyguard knights. Bedouin Camel Warriors also tend to be good generals and governors. Egypt gets a discount on camels, too.
Nubian spears are a good early unit, excellent for garrisons, but watch their loyalty.

3. Early building
Every province on the Middle Eastern coastline except the Sinai is a glorious achievement goal for someone's Crusade. Therefore, the first order of business is to get the facilities to make dhows, and then build two. One should cover the Nile Coast and the other the Eastern Mediterranean. This cuts off direct invasion by sea.
The second order of business is to get a spear-maker’s workshop somewhere to make Saracen Infantry. The combination of those and desert archers are extremely effective on the rabble in the first wave of Crusades.
Use Armenia to produce Armenian Heavy Cavalry with a bonus. Build up Syria for their assassins. The two-star bonus there makes these assassins extremely effective.

4. The Navy
Extend your navy northward, through the Black Sea. This prevents a back door Crusader landing in Georgia. Then go to the Ionian Sea and up into the Adriatic. This trade route is not dependent on the Byzantines. Antioch is a money tree once a trade network comes to life. Tripoli and Egypt are rich, too.
If you are not at war with the Byzantines by now, start one. Always keep the most important goal in mind: Constantinople. Constantinople’s developed fortress is the key to getting a Grand Mosque for Imans and Nizari. I always build the Grand Mosque there, even though the glorious achievement goals want it built in Egypt. Be sure to take Georgia and fortify it, of course.
Now you have a very nice empire, defended with three land chokepoints: Constantinople, Georgia and Egypt. Expand along the North African coast at the expense of the Almohads. You can push all the way to Cordoba and still end up at a single chokepoint.

5. Crusades
War with Crusades is constant. You should have enough trade wealth to afford plenty of troops, though. Most of the early Crusades will consist of rabble such as fanatics and other unarmored units. These are very vulnerable to desert archers.
The movement programming of the Crusades means that French, English and German ones must keep trying to march through Constantinople if you push other factions far enough back in North Africa. The option of a Crusade landing in Cyrencia, for instance, will not exist as long as you own Cyrencia and have fleets to defend it. The Spanish could Crusade through North Africa, so keep a sizable army there.
Your goal is not to beat Crusades in a fair fight, not unless the odds are extremely favorable. Your goal is to corner them and have them die slowly. The enemy faction cannot launch another Crusade while its first one is dying, not unless the Pope gives a special call.
If you own Constantinople, you simply block Crusades with a large army. If your are not in Constantinople yet, the goal is to keep your armies in front of the Crusaders and to block them, keeping the restrictive movement rules for Crusades in mind.

6. Jihads
Build a ribat, which provides a good morale boost, and a Jihad in every province. Jihads are very effective, particularly in rebellions and civil wars.
That gets us out of the Early Era. Keep fortifying Georgia during this time, for the Golden Horde is coming. Also, build with Faris and Mamelukes in mind.

EDITED: To include the summery compiled by Doug-Thompson