EARLY
The Empire is one of the least stable of the factions. It starts out with many low-loyalty generals spread over many provinces that are in contact with a half-dozen rival factions.

My first recommendation would be to simply forget the early era Crusader goals. The Empire is not stable enough to survive a failed Crusade, and you cannot spare enough good troops from Europe to make sure the Crusades succeed.

I like early blitz openings, but even I advocate a build-up approach with the HRE. Build that farm economy and some defenses. The good steward and builder virtues will increase loyalty and happiness, greatly improving stability. I would also highly recommend managing your own taxes and keeping them low to further increase loyalty.

Players that are more adventurous prefer to blitz the Italians, taking all their continental provinces. Then they attack the Pope when they can bear the excommunication, take Rome and Naples and reduced Papal armies to a small remnant cornered in the Papal States. That satisfied the “Holy Roman Empire” goal.

You start with one province -- Bohemia -- that can produce Slav Warriors from the start and can produce Slav javelins once you add a spear maker. These are excellent units for the price. The rebel province of Pomerania can be overrun, and they can produce the same units. Peasants from Provence get a valor bonus.

Exploit rebellions and civil wars. Especially look for any chance to take over Denmark. Rebellion, inheritance and even a quick blitz attack if you can pull it off without excommunication are all options.
Denmark is particularly important for two reasons. It can produce Viking huscarls in the early era. These superb units can be the key to successfully defending the whole empire. Second, Denmark is the land bridge to Sweden. A HRE player who takes Denmark and Sweden relatively early is a happy HRE player.

The Danes produce many heirs, for some reason. Expect their army to have many Royal Knights if you do not attack them early. This is one of the reasons I like Slav javelins. Javelins are great for taking out knights, which would normally slaughter low-tech infantry. They do take some skill and practice to use, however.

If you do not know the details on how excommunication works and how to avoid it, learn. The most important point is that you only get one warning at a time. For instance, suppose that at some point in the game you attack the Polish and get a warning. You go on the defensive with the Poles and attack the Hungarians. You are now free to keep attacking the Hungarians.

Keep track of the year when you get a warning. After the Pope's time limit is up, you can attack the Poles again, in this example.


EDITED: To include the summery compiled by Doug-Thompson.