Draw 3 is almost always the most powerful of the benefits. The other two vary depending on exactly what kind of deck you're playing, and what point of the game you're in. But opponent drawing one is also significantly the worst of the drawbacks for you, and it is not to be underestimated.
Draw 3 on the non-terminal is like playing 2 Labs in terms of what it does to your hand, while an opponent drawing a card is like them playing a Lab. Net, you're getting a Lab effect, very roughly. Of course, there are questions of whether the returns from this are diminishing or compounding, as to which is more valuable. In general, Lab is a fine but not great card. This option is, of course, heightened when your opponent can't take advantage of their cards in a meaningful way.
Gaining a gold is sometimes fine, occasionally good, sometimes mediocre, rarely bad. However, what we're really looking at here is the difference between Gold and Silver. Maximizing that difference usually happens when cards (drawing) are at a premium. Also very relevant here is that the cost difference can be a boon with trash for benefit cards - including the third mode of Governor.
The Remodel option is often clearly the worst. Non-terminal remodel can be decent, of course, but you really need good targets, and often giving the opponent a free Upgrade is quite bad, especially early in the game. They can (usually) get rid of a copper off of this option, whereas you can't. So in the early to midgame, you want to pick your spots - 3s into 5s are often quite good, and Estate into 4 works as well IF there's a particularly good 4 for your deck.
Governor is clearly very good when there's a discard attack available (though watch out for Moat or Lighthouse), or Possession, etc. Just take the cards and be very happy at your hyper-efficient draw card. There is some depth to these situations, but I'm not going to cover them here.
"The Governor Deck"
The classic Governor strategy, and one which some players seem to think monolithic (and strong and thus boring) is to A) get as many governors as possible, B)draw a bunch of cards and gain some golds, then C) draw a zillion cards, Remodel Golds into Provinces, win the game. Another problem people have with this is that it can be quite coin-flippy as to who wins, based on whoever gets that one turn at the right moment.
However, I think this deck is both not so strong, at least without help, on the one hand (though not really weak by any stretch), but more importantly, consistently mis-played. In general, I find that people choose to draw far too often on this plan. I almost never choose to draw until I think it's going to be the last relevant turn of the game. When you do reach such a turn, it is of course safe to draw, as they can't take advantage of the extra cards if they don't have a turn to use them in.
Let's break down why I think drawing too much is a trap, even though in general Lab is probably a better card than "Gain a gold, each opponent gains a silver, +1 action". With this particular deck, when I am gaining gold, and buying something good, my deck is improving pretty significantly. Later in the game, I flip to straight remodelling, which makes my deck worse, and theirs a little better, to be fair, but with the explosiveness that is possible, the points are pretty significant. Furthermore, the silvers tend not to be so great for the opponent. Gold isn't super hot, either, but the big leg up it has is that it has good applications from merely costing 6.
Generally, drawing means you are playing for a big turn. This is going to often require multi-governors, and things to remodel. This means a lot of cards, so you need to draw a good amount. Usually, you end up spending a couple Governors to do so. If I play that way, then my opponent will have more cards in their hand to start, which means they need to spend less of their Governors for draw, which means they have more in reserve to spend on other things. That's a big game for them. Instead, if I sit there and just gain gold or remodel, now they need to use more Governors to draw, which means less to remodel with. The silvers also make it harder for them to line up Gold-Gold-Gov-Gov - it just takes so many cards. Whereas if they are going to try to draw to do this, I have pretty reasonable chances of just having Governor to remodel and be able to buy a second Province anyway.
A massive skill in these games comes in trying to judge when you need to "go for it" and try to get your big multi-province turn off, and when to punt it down the line. The things to keep your eye on are the count of Governors and of 6-costs (traditionally Gold, but something like Border Village is very very good) in both decks, whether they've yet been seen this shuffle, total cards in both decks, and the Province pile. In general, I don't like making a move until you're likely to be getting enough provinces to either end the game or that your opponent can't realistically come back in one turn - keeping in mind that these decks tend to be pretty bad at getting multiple of any green card other than Province (though your mileage may, of course, vary). Every time you draw rather than something else, you give half a Governor draw back to them. Given that they draw less, that means you're drawing half again less, too. This effect compounds (though obviously there is a limit). So over the long run, you're leaking less value to them. The other benefit of gaining Gold is that, after a while, all those silvers tend to clog them up much more than your Golds (which are especially good because you want to remodel them).
Earlier in the game, drawing can be a little better, but you have to be very careful, very tactical. You want to be getting a tangible advantage from it without giving them a commensurate advantage. Particularly, you're usually looking to maximize the 5s in your deck (specifically Governor, at the least). So if drawing is likely to get you more 5s without doing the same for your opponent, then go for it. Typical ways to do this are looking to remodel a silver into a 5 and buy another. You really need to be aware of what you're likely to draw, and have a decent understanding of your opponent's deck, too. If you're really on top of things, you can sometimes mess the opponent up by triggering bad shuffles for them - this is a level above what I see almost no one be able to do, at least on a consistent basis, though. This whole line of play tends to close out when the Governors run, though, since that's when silver starts to get worse and gold get better.
Improvements on the Governor Deck
A straight Governor deck can actually be reasonably clunky and end up playing a bit Money-like. It's significantly better if you can engine-ize it. So get your trashing going, get other sources of draw going. Then you are free to use your governor as payload, gaining Golds and then remodelling them, and also perhaps a little bit of supplemental draw/stability insurance. Yes, this is good even if it means getting fewer Governors. In all honesty, you don't need so many Governors if you go this route, and though you obviously would like as many as you can get, it's not as important as the other stuff. Cards like Upgrade, Junk Dealer, Stables, Wharf, etc. get priority, at least for a while. Lab is probably not good enough to be on that list (though you will want Labs once the Governors run out) - just to give a general impression.
- If your opponent can't use the extra cards because they somehow only have 1 gain, you're much free-er to draw, as the worst they can do is 1 Province.
- Governor is significantly weakened by Colony, just because you can't remodel into it - this makes the gaining Gold a lot weaker, which in turn makes the whole card a lot closer to Lab, which is fine but not stellar.
- You can beat Governor by going for Big Money, but it needs to be very good Big Money, as you're letting them off the look in terms of drawing quite a bit.
- Be cognizant of the opponent being able to upgrade into points. This means changing the order in which you do your Governor options sometimes. Sometimes that means Remodeling then drawing (though that can be risky because it cuts you off from draw+draw into extra Governors, but track your deck and know what you need). Sometimes it means Remodeling into something to SEE if they Upgrade, and then being ready to buy (or not). If you're talking about the last VP card in a pile, knock it out on your first Remodel, getting your other points later.
- Usually you don't want to buy different cards just to be able to Upgrade into them - it isn't worth it. Just be aware of the different VP cards out there at different costs.
- Drawing is also better against Draw-to-X. The other options (especially Remodel) can be used WITH draw-to-X.
Game Log Examples
This game is a good example of how building up an engine, with trashing and non-Governor draw, can really get the job done, even in lieu of many Governors.
In this game, there isn't much other engine besides Governor (potentially Minion). However, I'm able to win despite losing the split 7-3 because of how the game plays out (and good luck on draws after the point the Governors run out). Big point here is that my opponent greens too soon, and is left with too little economy left to finish the game out.
Another game that shows how building a classic engine can be extremely effective. I get a Junk Dealer to thin down and supplement with Stables before moving in on the Governor to close things out.
In this game, an engine built with Governor supplement is fast enough to defeat even a Rebuilding player.
An excellent demonstration of principles. After some mutual ambassadoring, we both build. I am able to get a more consistent engine, and even though the raw power of Governor allows my opponent to still get 4 Provinces quickly, I'm always on top of things, maintaining high quality of deck to be able to put it away at leisure.