Events increase the strategic and especially the tactical complexity of Dominion: now you don't just need to know what to do with your cards and add to your deck every turn, you have to worry about what to do with the game-universe too. Here's some tips on each event to help you. If the only piece of general strategy can be "it depends on the kingdom," the only piece of general tactics is "pay attention." Every buy phase, spend 2 seconds thinking about events. It's not 2010 anymore.
The constant vigilance events: Borrow, Scouting Party, Save-- if any of these three are out, you are going to have to think hard about them every single turn. Borrow is expensive but often worth it, especially in the late game. Scouting Party's free buy makes it an auto-buy on many Province hands. Save should be used virtually every turn that you have +1 to spend more than you need, and should occasionally encourage you to buy cheaper, just so you can save. Don't forget to save junk cards from missing the shuffle, to save Reactions, etc. Obvious but easy to screw up: Saving a copper effectively costs two, saving an action effectively costs one.
Honorable mention in this category to Travelling Fair (which is of the same caliber of constantly-possibly-useful as the others, but often too expensive). Can't mention travelling fair without mentioning the insane Counting House Combo, of course, or that it can provide the only +buy on a board. Honorable mention to all the "gainers" as well.
The game warper: Inheritance. Inheritance, inheritance, inheritance. Inheritance. Perhaps the most fun card of the set, maybe the whole game, certainly the one that least makes you feel like you're playing the same old game of Dominion. If this is on the board, ask yourself: how can I hit seven as fast as possible? The laws of Dominion have changed. It can make sense to open silver/silver on a strong engine board if a sufficiently amazing target for Inheritance is around. Indiscriminate trashers are now mediocre openers, Moneylender, on the other hand, is good again. Oh, right, and the weird mid-game rush for the estate pile. The estate pile! The mid-game!
The steroids : Mission, Expedition. Mission tends to alternate between broken good and virtually useless. Sometimes you buy it in the early game just to play an important attack/traveler/etc, but often the early game is when you can least afford not to buy anything. Expedition, on the other hand, serves two very different functions: the obvious one (helping you get a megaturn or just a good early turn when you don't particularly want another Silver -- note there's no one per turn limit), but also the safety; if you're ahead in a Village/Smithy type game, an Expedition is probably better than an extra piece in the end-game to stave off a dud turn.
The cards that modify other cards : Gold medals to Lost Arts and Pathfinding. If you're not a fan of the Variants and Fan Cards subforum, consider going there briefly and reading some of the common arguments about how to cost cards, the non-linearity of card costing, the foolishness of adding +1 Action to a middling fan card just to make someone buy it. Then note that you can break all those rules, while literally playing Dominion, with these events. Festival + Pathfinding = Grand Market. Smithy + Lost Arts = unpriceable broken good. And yet Mountebank + Lost Arts = eh, when were you planning on drawing two Mountebanks together anyway, and did you really have nothing better to spend 6 on in a Mountebank game?
Silver Medal to Ferry and Training: Ferry's certainly also nuts. Not as game-breaking as you fantasize it to be, typically because the sorts of games where Ferry is unspeakably awesome are the sorts of games you could do something unspeakably awesome anyway, and Ferry just speeds it up. Training is still good, but it needs to go on something you're regularly playing more than one of per turn to be worth it.
Seaway, like many +Buy cards, is extremely important if it's the only +Buy card on the board, but is rather ignorable otherwise. Bronze medal.
The trashers : Bonfire, Plan, Trade. Bonfire in the early game is a little like a Mint open: good if you have some way to get treasures into an otherwise wretched deck, bad otherwise. Quite synergistic with the Page line. In the mid to late game it can trash silvers for you, if it's that sort of game.
Plan. Eh. Plan's useful in the early game the way Doctor overpay can be, but often gets useless pretty quickly. You find yourself buying a Shanty Town with a spare buy turn 12 and as the client asks, pathetically, if you want to trash a card in your hand, you remember, oh right, I had such big Plans back in the day.
Trade, like the card Trading Post, is at its best when there's handsize increasers and/or cursing. The comparison to Trading Post stops there; it's more expensive in the sense that you only have to buy Trading Post once, but it can gain you more Silvers, it can't cause you a disgusting turn 6 when you draw it with the only other good cards in your deck, and it isn't hanging out in your deck once you don't need it anymore.
The gainers: Quest, Alms, Pilgrimage, Ball. Quest you either totally ignore or you buy it almost every other turn, the latter in sloggy games where the two curse rule or an obsolete Sea Hag quickly takes you to gold-town. It's quite useful in Mercenary games as well; if your Urchins don't line up on the second shuffle, trade the second one for a gold; later it gives you somehting to do with a Mercenary you can't afford to play (or that you draw dead). And it gives yet another reason to consider Ambassador/Ambassador or Swindler/Swindler openings.
Alms, of all these, should maybe go in the constant vigilance category. The one-per-turn limit means it's never truly insane, but you still often give up a free Caravan if you fail to pay attention to this card throughout the game. Obviously best (not counting the early game) with money-producing-actions and +buy. Don't forget to consider Alms-ing for Estate/Tunnel/Island on an end-game dud turn. Also consider splitting 5/2. If you split 4/3 there's always the "I'll use my coin token" joke. The sequel to that joke is "I'll Alms for a Caravan" when neither is on the board and you don't want anything; you don't have to cite me, you can just use that.
Pilgrimage requires planning; you'll want to have bought the first one before your first mega-turn, but buying thin air in turn delays the average onset of that mega-turn, which can itself snowball. Most important when there's no +buy, or when there are annoying-to-gain cards like Scrying Pool or King's Court. The decision calculus changes with Ranger/Giant on the board, but there's nothing smart to say there except, you know, know which way your Journey token is flipped.
Ball, hmm, it's sort of like Band of Misfits -- it's fine, it seems at first like it's going to be great, but it just can't be because it makes you gain 4s when you have 5 to spend and usually 5s are more important than 4s. The -1 token is a real kick in the pants too given that you're already a little sad to spend 5 on a 4. I mean remember that time you opened Talisman and then you drew it with 4 coppers on turn 3? Wasn't that already sad enough?
The attack: Raid. It's expensive and floods your deck with Silver, which means it's only a good buy if you're willing to commit to buying it a lot. But then the attack is quite strong, comparable to Minion. Problem: do you know what type of deck is extremely vulnerable to a Minion attack? One whose economy is based on Silver-flooding. The Raid-spam, and especially the Raid-spam mirror, is a bit of a psychological game too: gain a Province now? But then you're not attacking, and you're not gaining 4 silvers. Well, does the opponent look like someone whose next hand has 8 money in it?
