Welcome to the forums! Here are my thoughts, which are not guaranteed to be that great.
In game 1, you should not be too focused on Trusty Steed/Feodum. While that
can be profitable, the problem is that it is unreliable. You cannot guarantee that you'll be able to connect Province and Tournament, and even if you do, it might be too late (whether the game is almost over or your opponent gets the Steed first).
You should be trashing more aggressively. On turn 4, you discard Overgrown Estate. It would have been better to keep it in your hand to trash with JoaT, which would have drawn you a fifth card anyway. On turn 5, you should trash with Steward and just not buy anything. Trashing is extremely powerful and Steward is a good trasher.
Overall strategy on this board is actually going to be pretty tricky though. Strong trashing makes Tournament a consideration. On the flip side, Knights could wreak havoc against a tight deck. Cultist and IGG are strong junkers too. I think the game will have to be very reactive. My general strategy would be to trash down and pick up some Tournaments for early economy and Hamlets to facilitate an engine, then pick up Cultists and Knights to slam my opponent.
But again, it's tricky. I'll leave it to better players to suggest more specific strategies. My main advice is just to either trash more aggressively with Steward. If you're not going to trash aggressively, you're better off skipping Steward and doing something else. JoaT-BM? Rush the IGGs? I don't know, but Steward is best for the early trashing, with the card and money options only minor bonuses for after trashing is done.
Game 2 is definitely a Duke kind of board. I think I would skip Cache here. Cache costs $5, which is much better spent on Duchies and Dukes. While the Coppers don't hurt, they slow you down. Even on $5 I think I'd rather take Haggler. But the real star here is Horse Traders, which gets you to $5 very reliably. There's even an article about
HT-Duke!
I think I would open HT/Candlestick Maker on 4/3, Haggler/CM on 5/2. I like CM here because it is non-terminal and the coin tokens will allow you to green even earlier and still hit $5 consistently. Haggler is nice so that you can pick up Duchy/HT or HT/Silver or HT/CM. It's just pretty flexible. After the opening, I'd try to get two HTs to start and then immediately go after Duchies, maybe picking up a third HT on a $4 hand. In a mirror, you need to get all the Duchies you can before they are gone. In a non-mirror... I'd still go for Duchies first most of the time. Taking the Duchy denies those possible points from the opponent.
Looking at your deck composition at the end of the game, I'd say that you definitely overdid it on Cache and you could have used a few Candlestick Makers. Cache did you much more harm than good, because every time you bought Cache, you could have bought Duchy instead and ended the game that much sooner.
Game 3: I think I would go for Duke here as well. Sea Hag to start, and Haggler is decent support. One mistake you made was declining on Duchess so often. Your deck could have handled a lot more. Duchess is not that bad and it would have improved your deck.
One way to improve your play here would have been to track the score. If you can keep track of how many Duchies and Dukes you have, you would have known not to end the game when you did. Granted, your opponent could have ended the game on a win by buying a Curse on the next turn, if he'd been paying attention... but yeah.
By the way, your opponent's lead wasn't really insurmountable. If you look at the graph, the game was close all the way down.
Game 4: I think I'd go for Storeroom-Gardens. I would have opened double Storeroom and skipped Silver entirely. Every play of SR is guaranteed to give you enough for Gardens-Copper or Estate-Estate. Mountebank doesn't harm you.
Edit: Took another look at game 1 -- you
never trash any Copper? You have 8 at the end. Trash those Copper!
