This page deals with different expansion strategies that can be used in the game. Remember that none of these is an absolute one and there are many ways to accomplish things.
Radial expansion means focusing on capturing the provinces surrounding your home fortress. Since your home fortress drains a proportion of your surrounding provinces resources this strategy can work well for resource-heavy nations that have strong troops.
For example it won't be too hard for Ulm to build up a force strong and diverse enough to deal with most independent types and Ulm will benefit from the resources you can gain from the surrounding provinces.
However this strategy has its drawbacks. First of all, the provinces you start next to won't always be profitable ones. You can start next to three plains and two swamp provinces which provide only small amounts of resources. Sometimes the independents you start next to are too hard to defeat for one to pursue this strategy well. You can start next to two provinces filled with Knights and Longbowmen, one province with Blood Vines and one lightly defended one.
Perhaps the most optimal starting position for this strategy is starting next to several Mountain and Forest provinces.
After the initial phase of capturing the provinces around your starting fort you should start to do long term planning. Will you go after lucrative provinces, or will you plan on find a spot for your next castle? If you plan to set up your next castle soon you should scan the map for a suitable nearby province and then head to capture it. Start building the fortress if you have the money and then carve its surroundings. Repeat as many times as you want or then switch your strategy if situation calls for it.
Tunneling means going after the most lightly defended provinces. Since the more provinces you have the more income you draw, the better it will be.
This is perhaps the easiest expansion strategy to pursue since pretty much everyone can mass forces and pick a weak enough target. Nations with fast strategic movement will however have the easiest time to supply reinforcements if something goes wrong.
The downside of this strategy is that this can result in empires that are hard to defend against enemies. If the "tunnel" is too long, it is possible that an enemy attack can sever it and it is harder to defend a narrow territory in general. The easiest provinces are usually the poorer and insignificant ones so your income might be lagging behind more focused empires.
Tunneling can however result in fast territory expansion. You must however keep an eye on your empire to make sure you will not make it undefendable and too narrow. You should try to build up task forces to take stronger provinces for you, especially the ones that surround your home fortress.
Shotgunning means using flying or sneaking units to pick poorly defended provinces, attacking and then moving on. This will result in "shotgunned," decentralized empires. This can work for your or against you.
Pangaea and Caelum are perhaps the best ones in this strategy. Pangaea has many different types of sneaking troops and Caelum has flying troops and more importantly flying scouts and sneaking priests in the middle era. Scouting is important for this strategy as you need to pick out provinces you can overpower.
One good side is that this can allow you to find a spot for your second castle very soon. This can also allow you to pick valuable provinces in some cases. Picking the three Mountain provinces somewhat near to you as Caelum can give you fine access to fortification spots and possibly even magic resources.
The downside is the high decentralization. You will quite likely not have as many provinces as other empires and your empire might have less build capacity than other.
However this decentralization can also work for you. It will be very confusing to go after an empire whose 10 provinces aren't in one "blob" but scattered around. In the best case you can launch harassing attacks against the enemy rear echelon while the enemy has to fight its way through independent provinces while you plot a suitable battleground where you will attack from the shadows or from distance with overpowering force while you cut their escape with a smaller task force.
A somewhat risky strategy to counter enemy "tunnellers." The idea is that you do not start tunneling toward your opponents castle but instead wait for him. Conquer the provinces around your home fortress and maybe some more provinces. Maybe build more fortresses. Build up big armies of good troops (like Jotunheim giants). Do not conquer tough indy provinces (Knights, Longbowmen), instead wait behind those provinces. If your enemy moves toward your fortress he must first fight those tough indy provinces. Now he is most likely weakened and you can overrun his army and move in his "tunnel" of conquered provinces direct to his fortress.