Are there any quick start guides around here for Shogun 2 (Fall of Samurai specifically)?
I tried getting into a campaign but was overwhelmed and the tutorial didn't help much. In particular I have no clue how trade works.
This is always an issue with TW games. In Empire and Medieval you start with four provinces and several armies to manage right at the start - Shogun 2 is pretty kind by comparison (particularly since province management has been heavily simplified and you're no longer presented with a dozen different buildings each with its own upgrade path from the start of the game), but the best tutorial for Fall of the Samurai is actually Empire's US independence campaign, since it introduces modern warfare!
Yes, trade is simple - there's nothing to it beyond clicking 'Request trade agreement" and everything else is automated. The supply/demand/cost per resource etc. screens appear to be there simply for flavour (some strategic resources are available through trade, but you tend to need strategic resources only late in the game by which time you'll probably have sources of most yourself).
Total War campaigns play more simply than they look; they have more of a learning threshold than a learning curve, in that once you overcome the initial difficulty of working out the basic system everything else on the strategic side tends to be rinse-and-repeat. Ultimately they boil down to the usual strategy game formula: make money, then build army, then go on rampage. The guy with the strongest economy wins. Managing money comes down mostly to managing taxes and happiness - taxes make people unhappy, repression makes them happy again. Farms make money and food; castles that repress people eat food. All other building decisions are optional to some degree, and most are independent of one another (although certain units require two or more - stables aren't much use without a dojo for producing bow, spear and/or sword units, and Warrior Monks can only be recruited if the province with a monastery also has spear or bow buildings).
Shogun 2 is much simpler than the older games in this regard; those had a separate population mechanic to manage, which determined how fast you could upgrade your towns, and with its own building chains to manage; in Shogun you can upgrade anything with enough money. You had many more building options, and you no longer need to create or develop agent characters for diplomacy or research, or use generals as provincial governors (with a whole series of related stats). Unit replenishment has also been simplified and automated - you don't need specific tech buildings for units to replenish, nor do they need manual retraining (on the downside they can't actually get manual retraining if you want it, say if they're in a province with an armourer).
Don't over-rely on trade, because although it makes you money quickly - and in the early game, securely - the eventual 'realm divide' will ultimately cost you all your trade partners, and before that make you vulnerable to having your trade routes attacked, giving you the unpleasant surprise of suddenly going bankrupt when you have a giant army to support.
The best approach is to start with a few turns following the missions you're given, as at least in the tutorial Chosokabe campaign they're intended to introduce you to game concepts (you'll usually get a 'Beat this army' mission, then a 'Take this province' mission, then a 'Use an agent ability' or 'Build this building/unit' mission; in both Rise and Fall of the Samurai, but not that I've noticed in vanilla, you sometimes get 'Enter diplomatic agreement with Clan X' missions as well).
You then need to train yourself not to think of missions as a straitjacket - you won't get missions telling you everything you need to do; once you've learned to do X, you can do it again as you see fit.