The Mori clan starts off in the western part of Japan. Its not the greatest of starting positions with 3 enemy clans on its borders (Takeda, Shimazu, Oda) as well as a rebel contingent to the south. The Takeda are stuck right to your underside to start as well making it an immediate threat to all your territories.
Koku is scarce for the Mori clan to start. The region is not good for farming so the average yield from a territory is under 200 Koku. Once the starting Koku allowance is used up Mori is in trouble. They do have Silver deposits in Harima so that adds 400 koku. Also Mori can build a port cheaper in Bizen, which adds 200 koku to their income.
One strength the Mori clan possesses is the abundance of Iron and Sand deposits in their realm. There are 6 and 3 more along its immediate borders. This allows for the training of much more heavily armored troops, which will last longer in battle.
To help out Lord Mori's war effort the clan can recruit and maintain the best foot soldier in the game, Warrior Monks, for cheaper than other clans. Warrior Monks can take out any of the ground troops in the game with little to no problem. They also strike fear into the other Buddhists in the game since it is looked down upon in the religion to kill a Monk.
The Mori clan starts with 2 very bad things working against them. One is the Koku production of their lands. The poor income is not enough to maintain the army needed to even defend you existing territories. The second problem is the Takeda clan has 3 territories right in the middle of your empire. These 3 territories force you to post a garrison in almost every territory you own putting an even bigger strain on your economy.
Strategy:
Always start by taking care of the negatives that face your clan. The 2 problems stated above can be killed with one stone. Take out the 3 Takeda held lands in your realm in the first season! Yes, in the first season they will not have time to reinforce or attack you. First, take an Archer unit from Suo and a Yari unit from Iwami and attack Aki. This is the key battle to win. If you take out Aki, Takeda loses a port so reinforcements will be impossible from the East. You also take the castle they have in the west preventing troop production. In the same round, take 2 Yari Samurai from Izumo and attack Bingo. This will prevent them from diverting forces from Bingo to help in Aki. Then take your forces in Mimasaka and Hoki and attack Bitchu. This will keep Takeda from moving reinforcements into Bitchu to help that attack. The first battle is a must win in Aki. Once you win that you can fight the battle in Bingo. Should be a fairly easy win, if not don't worry. There will be no support coming to these territories so you can always go back at them next round. The third fight is a river. This is not easy with the small force you have, but Takeda has only one Yari Samurai unit there so fight it out. You can shoot him with arrows, if he's out of range; take your Yari Samurai right at him. Then attack hand to hand with your Samurai Archers. That battles over!
Now you have taken care of your starting koku problem by adding 600 koku to your income. You have also taken one of your enemies out of the way. Now you have Shimazu in the west and Oda in the east. You also do not need an army in all your provinces now to defend yourself. You can concentrate them at the borders and start building.
Your initial solidifying strategy is not done however. It is wise to never fight on two fronts at once. Since Shimazu usually keeps his alliances and is busy fighting Imagawa its best to ally with him and expand into the east. Besides, there is more koku in the east then the west. So send your emissary down to find Lord Shimazu and make peace with him, but before you do take Nagato from him. You will then control the entry point to the mainland. Shimazu will no longer have a foothold on your soil.
Don't Get Comfortable:
Now you have added some koku to your income and gained some land don't get comfortable. The Mori clan income is still very dismal in comparison to some clans. You need to build up some good units now and prepare for the offensive.
First things first, get that Buddhist Temple up in Mimasaka. Your clan specialty is Warrior Monks so don't be shy using them. You don't need to swamp your army with them but in a 16-unit army, 6 or so units of monks doesn't hurt.
The next production change you will want to make is to move the building of Yari and spear units to Aki. I prefer to build troops where here is a port if I can because it makes moving them later on very easy. Aki has Iron and Sand deposits for an armory so you can also build Naginata here.
Don't wait long to attack your enemy. You will run out of money and not survive. You must expand east to gain more money and build a more devastating army.
Where To?
Picking the proper territories to attack first is a tricky art that takes experience to perfect. Especially in the case of the Mori clan, you can expand east west or south. However, this early in the game the rebels in the south do not have their hearts set on expansion so you are safe from them. The Shimazu territory is very tempting target because it is a single island and by taking it you would have the entire west side of the map. However, this strategy is flawed. The Shimazu lands are almost as bad as your own lands when it comes to koku output. In the meantime your enemies in the east are stock piling the massive koku they produce and building really big armies. So the choice is to expand east.
Tajima is a good target first. It is controlled by a small group of rebels and can be taken fairly easily. This puts you flush against the Oda clan. One of the bonuses of the Mori clan when advancing to the east is the option presented to them in deciding which provinces to take first.
When in doubt take the province with the highest koku yield. That way you not only add a lot to your own but you also take away from your enemy. Try taking Kawachi, it has the highest Koku output of the adjoining territories.
As I said the options from there are limitless. But keep in mind that Kii is near by. Although the koku output is low the territory is famed for its Warrior Monks, which receive a +1 honor bonus if produced there. So get Kii and move your Warrior Monk production there.
A note about Yamashiro: it is a very tempting target due to the honor bonus to all troops produced there. But be careful, the standing army there is very, very powerful. It's mostly made up of honor 4 Monks and Archers. Your best bet is to develop some Cavalry before you take it. Also keep in mind that the rebels there are not very expansion minded so they don't pose as much of a threat as Oda does.
Although I haven't mentioned them much here, special units should not be ignored. The Mori clan doesn't need to produce a lot of these but shinobi are invaluable when conquering new lands and for scouting out new lands. Ninja come in handy when you are having trouble beating that certain general on the battlefield, you can assassinate him. Tajima produces Shinobi with a +1 honor bonus. So you can start a shinobi/ninja producing facility there for your special units.
The situation around you at this point may be different than what it was for me so it's hard to say what to do next. Keep these points in mind.