General Tips[]
Opening moves aside, Shimazu end up looking a lot like the Otomo: controlling Kyushu and getting rich off trade.
Shimazu should consider going Christian in the early-mid game, to access the better church buildings and research rates, as well as Nanban trade ships to dominate the seas. When their daimyo eventually dies, they can go back to Buddhism to access bow warrior monks.
Starting Position[]
The Shimazu start in Satsuma and should go for Osumi ASAP. The Shimazu should build ports and ships as soon as possible, to occupy nodes. If they're fast they can grab the four nodes surrounding Kyushu--but even if you rush for the one north of Honshu you won't make it. From Osumi and Satsuma should then try to take Kyushu. The mid game then transitions to much like the Otomo's.
Research[]
Shimazu have two outstanding unit types: Heavy Gunners and katana samurai. Beyond that, Chonindo for maxing out bow warrior monks and shimazu heavy gunners is good. Focus on the military technologies after the basic eco techs are done: after converting to Christianity, getting the monks, and getting the church buildings, you can switch back to eco techs for the faster research rate. Finally, Shimazu is one of the only factions that can consistently get access to geisha, so ninjutsu mastery is a worthy go to.
Army Comp[]
Depending on unit settings, two to four heavy gunners can cover the entire front of an army. Support them from behind with 3-5 units of bow warrior monks. The combination of the warrior monks' deadly volleys and the shimazu's demoralizing and havoc-wreaking bullets can destroy armies before they even reach the front lines. Should they somehow reach the lines, however, 4-6 units of Shimazu's superior katana samurai should mop up any survivors. Have 2-4 units of yari cav deal with any flanking infantry or cavalry. 2 naginata samurai can act as distractions and soak up enemy counter fire; after the archers are dead, they retreat to the flanks to fend off cavalry.
Turn 1[]
The Shimazu have *ostensibly* the easiest position in the game, but I've found that, depending on diplomacy, you can get royally screwed. This order of actions gives the best possible outcome from what I've found.
Besiege Osumi's capital with just your daimyo. Use your other army to chase the nearby Ito army away, then take Osumi. Do it manually if you want to save a bit of money; repair if you do it automatically.
Build a harbor in Satsuma, and train three ashigaru in Satsuma. Research Way of Chi. End turn.
Turn 2[]
The Ito army will move south and raze the harbor in Osumi. Send your secondary general and the initial army south to kill it. Move your three yari ashigaru from Satsuma to the capital of Osumi with your general.
Ask Sagara for 1000 koku in return for 5 turns of access. Use the money to build a harbor in Osumi. Exempt taxes in Osumi. Any remaining money should be spent on farms.
Turn 3 and onward[]
The next Ito army will attack Osumi. It should be an easy siege win, and if you play your cards right you'll get a heroic victory.
From here on, you can ask Ito peace and make them your vassal. Hyuga's a terrible province so taking it won't benefit you much, and it takes forever to get there besides. Or you could take Hyuga eventually, preventing the Ito from backstabbing you later.
Ask Sagara for more money in exchange for 10 turns of access. Then ask Sagara for more money for 20 turns the next turn. Build ships to occupy all the nodes.
Very often, Sagara will get killed by the Otomo, who may or may not declare war on you and/or the Ito if you've vassalized Ito. Use this opportunity to take Higo off their hands. If you've vassalized Ito, Ito may do the same to Bungo. That will leave you with the Shoni to the north, who usually takes Buzen and now has three provinces.
If you are unlucky, Ito will backstab you and the Shoni will declare war on you at the same time. If you're lucky and only one of these things happens, or it happens in a fortuitous way (Shoni and Ito killing each other, for example) then the game is as good as won.