Clan Shimazu starts of in the southwestern part of Japan. There backs are against the water, which makes for a great starting position.
Koku is a problem to start with for Shimazu. The annual income is only 1194 koku in an average year. This is one of the poorest income levels of any clan. But there is help near by.
Clan Shimazu has only 2 real enemies on its borders. It has 3 Imagawa held territories in the north and Mori to its northeastern side. The rebel held island is of no threat at the beginning of a campaign as Ronin are not very expansion minded. Also the capital province of Satsuma along with Hyuga and Osumi are completely sealed off from enemy lines. This is a tremendous boost since the starting army is not all that huge.
To further help the Shimazu quest for shogun they can recruit and maintain No-Dachi Samurai for cheaper than the other clans. This gives them a punch that can prove to be a turning point in any battle for expansion.
Now the only pitfall of this clan is that it has a very low starting annual income. This can cost you in the beginning because if Mori or Imagawa decide to concentrate their forces and expand into your territory you not only lose some of that valuable Koku input but your also going lose some of your initially small army and open up lines to get to your 3 undefended territories.
Strategy:
First and foremost always take care of the negative aspects facing your clan at the start. For Shimazu the answer is easy, get more Koku. But How? Attack Imagawa as soon as you can. By taking out Chikugo as soon as possible you can add a 300+ Koku province to your income. That can provide a tremendous boost to troop output. At this time you should be building archers and samurai in your capital and spending money on nothing else. You don't have any need to tech up right now and even if you had a need you wouldn't have the Koku. So the next target would be Hizen. The reason is that it has a port and is being supplied with troops from the eastern Imagawa territories. Once you take Hizen you now have added almost 1000 Koku to your income. And in the process you have cut off Chikuzen from any troop support so they are just sitting ducks. And you can take them out at your leisure at this point. So if you need to rest and build some troops do so now because you have bigger things to tend to soon after you take Chikuzen.
Now you have exactly 2 borders to defend, Nagato and Bungo. Keep in mind that Bungo is not all that expansion minded but if you leave a territory undefended they will take it. So build up a decent force in Bungo and a major force in Nagato. Once you feel you have solidified your borders you need to tech up before your next leg of expansion.
What to Build?
First your empire is long and moving troops from where you build them is starting to become an arduous task. So ports are going to be needed in Satsuma and Nagato.
After the ports are built its time to build a superior force. By now you should have gotten the Legendary swordsman Event, if not don't worry it will be coming soon enough. When it does build the sword dojo in Satsuma. The No-Dachi get a +1 honor bonus in this province. In Nagato build up to Naginata. You can now move your Yari Samurai Production right to Nagato, which is your front line. So Nagato can supply you with Yari Samurai and Naginata while Satsuma provides No-Dachi and Samurai Archers. I find this combination the best.
Now is also the time for preparing your plans. You need to build Ninja and Shinobi to help out your war effort. Use the shinobi to guard your daimyo, you never know when there are some other ninja snooping around. Use the ninja to start looking around for you, getting an idea of what kind of troop build up your enemy has on your borders and behind his borders. This way you know what your facing.
Where To Now?
You will start to notice that keeping an army of your size is becoming too expensive for your Koku output. This means its time to expand. You have two choices here, you can expand into the Mori lands to the North or you can take a "backdoor" and go through the island held by the rebels. The only way to make an intelligent decision here is to use your ninjas and spy.
If you see that Mori or whoever has the lands to the north have little in the way of defense in the nearby provinces it might be advisable to go that route. However, if the enemy in that region is showing major force it might be better to go through the rebel island and "sandwich" them by coming out on the other side.
My bet would be to go through the rebel island and do it in a hurry. The forces there don't start out all that strong and when I played it was no different. By taking out the island you now have a defensive front in Nagato and one in Sanuki. At this point you will know what you are up against.
Start to expand from Nagato into Suo and the surrounding territories. If you notice that the bulk of troops are not being produce west of the Awaji area then move your force from the island and cut off the lands from the troop suppliers in the east. Now the troops and lands (old Mori) are isolated from help and can be worn down little by little.
If you notice that the troops are being supplied from within the Mori lands then do not move your troops from the island. Keep that army there holding the front. If you move from there and attempt to start attacking you will be faced with troops being produced on your left and right and you will be surrounded. This is not a good thing. When your army from Nagato takes out the main troop producing facilities on its quest across the old Mori land you can then move your army from the island and cut off the remaining territories. Once the two armies meet one another they can go on and conquer the west together.
At this point the game can go anyway it wants so its up to you from here on out. Just keep these few things in mind.