I've tried several strategies with Chokosabe at legendary difficulty.
You can either start off by conquering the whole of Shikoku or you can ally with the Myoshi in the East who seem to be pretty reliable allies if you let them take Sanuki.
The latter lets you immediately start expanding west -my favourite strategy is to land in Hizen in the far NW as that and the neighbouring province (which should be called Chikuzen but is spelt weirdly for some reason on the campaign map) are easily the two richest provinces on the western half of the map.
Having taken these you can then mop up the rest of Kyushu - and only then should you start on the many small poor provinces on the western end of the big island - taking Nagato first as it is the richest and then Suo and Aki as they are good for recruiting cavalry and warrior monks.
One problem with leaving the advance west too late is that the Ouchi, Otomo and Ito seem preprogrammed to go Christian quite early - so the later you leave it the more likely that the western provinces will be 100% Christian and thus harder to hold down.
(of course if you go Christian yourself that's a feature and not a bug - but personally I find warrior monks and the monk agent far more useful than matchlocks and missionaries).
You also need to focus heavily on trade as very few western provinces have fertile soil - and there are a lot of small poor provinces to garrison once you started rolling east down the big island - this is another good reason to take Hizen early as this is a lot better placed than Tosa or Iyo to spam trade ships to fill up the western nodes.
It is also important to establish trade links asap with the Hojo or whoever holds the richest eastern provinces - these should stay at peace with you a lot longer than the western and central clans.
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