Basically I agree with almost everything Mic said, I just need a lot more words for it.
I like to think about games of dominion as though it's about
1) an overall game plan
2) a turn-by-turn execution
For me, 1) bottles down to a simple question, represented by one of the users here: DoesItGoBoom?
On this particular set that would be the question of "can a deck with a lot of festivals and libraries beat a deck that has mostly money and just a couple of action cards".
After thinking about it the answer looks pretty clear to me - no, it's not going to work. Festival/Library isn't even that strong to begin with (except for with supreme trashing) and then there is this Witch ruining all constructive plans. So my plan here would be to mostly keep it simple and buy money and a couple of action cards.
If you reach that conclusion, it's important to be strict about it. Be *very* restrictive in the terminals you add to the deck. While all terminals provide some kind of a bonus over a money card, it's rarely worth it if you draw more terminals together. I think you did pretty well on that front except for the festival over gold. Festival is only better than gold if you draw it together with two other terminals, or if you can use the +buy. Using that +buy won't ever happen here, so really it's down to drawing 3 action cards together. That simply isn't worth the $1 it produces less. Not to mention the fact that it makes $3 less then a Gold if you draw it with Witch.
So... what action cards to select? Two witches, not 1 and not 3 - that is clear to me. I think I would skip Moneylender but buying one certainly isn't wrong. Getting a Gold before the 2nd witch is probably ok now that you're playing with Moneylender (3 terminals in such a small deck is probably a bit much), but also the reason I'm in favor of skipping Moneylender. Unlike what Mic says I'm not in favor of ever getting a Library over Gold, especially not if you skip Moneylender. The two extra cards have to make up for the $3 you're missing, I don't see that happen all that often. Library over Silver on $5 - yes that can be good. Now that you are playing with Moneylender, you could consider an Adventurer, which would be better than Library.
Remodel I probably won't buy at all. The only scenario where I want it is if we're already buying Duchies, I hit exactly $4 and I don't like estates yet. I think you correctly skipped all other action cards I didn't mention.
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About the execution, I think you're really off on the Duchy front. This is actually the only thing I think you did really wrong in this game, everything above is just marginal. When to start buying Duchies is complicated when someone is playing an engine, as it can have a major impact on the duration of the game. With these two relatively simple decks it's just about grabbing points. When you take a Gold over a Duchy, the reason you do so is because you hope that Gold is going to help you to Province. It might turn a $5 hand into a $8 hand. However, do note that at this moment you still don't have more points. You just have a different deck with Province+Gold instead of 2 Duchies.
So let's compare possible good outcomes of buying a Gold to possible bad ones:
upside: Gold+Province beats Duchy+Duchy if you see that Gold another time, and it allows you to buy more points at that time. This requires at least 2 shuffles after you buy the gold.
2nd upside: Maybe Duchies run out. Now Gold+Province certainly beats Duchy+Silver / Duchy+Estate
downside: Maybe that Gold you buy misses the shuffle and it requires 3 shuffles to see it twice.
downside: Maybe that Gold is one of the golds you have on turn 19, helping you from $8 to $11.
downside: Maybe that Gold helps you from $2-$4 to $5-$7. Now Gold+Duchy is actually less points than Duchy+Estate.
downside: having a point lead sometimes enables certain endgame strategies, like remodel Province->Province.
downside: having a point lead sometimes scares your opponent out of buying the 2nd-to-last province.
All in all I think your golds on turns 15 and 16 are bad, and the golds on turns 17 and 21 most certainly are bad.
I also think you played better on this board than your opponent did, and that you would have beaten him in a 6-game match easily.