I am extremely strict about playing etiquette IRL games.
Deck on the left, discard on the right, play area in the middle.
Play actions first, one at a time, announcing when they make you do something. It really doesn't take a long time to say "Peddler, draw, Peddler, draw, Village, draw, Smithy, draw three, Worker's Village, draw, Militia, Everyone discard down to three. I've got an action left, but no action card. That's $4, [play a Silver] $6, [play a Copper] $7, [play a Copper] $8, Horn of Plenty [counting the 8 unique cards] gaining a Province, trashing the Horn. Buying a Province. I've got an extra buy from Worker's Village, Peddlers are free, buy a Peddler. Cleaning up [scoop cards from table, placing leftover hand on them, putting all on top of purchased cards in discard pile]"
Modeling that helps. Even players in the group who have played a while start getting sloppy and flashing the hand, and I always, ALWAYS say, "show the money" and make them put it, face up, in the play area, where everyone can see it.
I have definitely had to tell many players to stop putting their Estates on the table. Those are not in play, they're in your hand. What's the difference? Ask Horn of Plenty, or Pilgrimage, or especially Haunted Woods.
Because IRL games can take a long time, I allow some leeway for shuffling. I first look at the kingdom and see if premature shuffling is going to be an issue. Is it possible in any way for any cards to affect other players' discard piles? If not, players are allowed to shuffle while the next person is taking their turn. In fact, if people holding draw-3 cards and their deck is down to two cards, I'm fine with them shuffling and putting the two cards on top while they wait for the other players to go. However, if it's relevant, I let them know at the very beginning and make sure that no one ever shuffles their discard until its their turn and they need to draw from an empty deck.
Gained cards definitely go straight into the discard, no exceptions. Again, if they say "what's the difference" I point out Watchtower, Nomad Camp, etc., and the fact that you can actually draw cards after gaining them if you gain them during your action phase. The fact that they go to the discard is extremely relevant.
Running action counting is encouraged along with left-to-right actions. Some builds tend to stack up a few actions before they can play terminals, or stack up a bunch of actions, such as Wandering Minstrel or City chains, so if players aren't alternating actions/draw/actions/draw, the players in my group have learned to stop for a moment and ponder "how many actions do I have left" before they play a terminal, and we count "You started with 1, [point to village] 2 [village] 3 [draw] 2 [village] 3 [Terminal silver] 2 [draw] 1...you have one action left."
One other critical one: Durations!
I insist on players always playing durations in a separate row on the left side of the playing area, next to the deck, even though that means they're not in order with the rest of the actions. It's still not too difficult to keep track of actions, because you really only need to count the total, not the order. The next turn, before playing, move any duration cards from the left to the right, next to the discard, saying aloud what they do and processing it: "Wharf fires, drawing 2. Caravan fires, drawing 1. Gear fires, putting these two cards under it in my hand. Fishing Village fires, giving an extra action." Naturally, it's hyper-critical that players leave these cards face-up in the play area and NOT in the discard. Again, "what's the difference?" You may have to shuffle this turn, and they're still in play, not discarded. And widely separating this turn's durations from last turn's durations prevents accidental-clean-up-syndrome ("Oh, wait, I cleaned up my Wharf! Nooooo! Can I fish it out of the discard?")
Wait, one more: Coin or no coin!
When playing a string of actions, many cards may or may not give coin, making it really tricky to count. Ironmonger, Pawn, Minion, etc. To indicate that a card has given coin (not necessary for Market, Peddler, etc) we push it up half a card, so the action row becomes jagged, with coin-generators sticking up.
You know, there are a bunch more, now that I think about it: Putting the -$1 token and guilds coin tokens in the play area where you put your treasures. Putting permanent durations, etc., (Prince, Hireling, Champion) above the deck. Putting out-of-play cards to the left of the deck (Island, Native Village. Or use the mat if you really like it.) When playing Storyteller, putting the treasures fanned out on the Storyteller itself, and leaving a little space after it to indicate you're starting from $0 from here to the right.
It might be temporarily irritating to keep needling people about playing properly, but eventually, it really smooths out the gameplay and makes it faster, not slower, because now everyone knows what's going on. So keep on insisting on using good form and following the rules.
Another: The card says "Reveal your hand." Reveal your hand, let's see it. The card says "Reveal a card from the top of your deck." Reveal it. The card says "look at the top two cards." You don't have to reveal them. The card says "reveal a hand with no actions." Show us. It's not that we don't trust you, it's that it might be helpful information for us to know that you had an extra Gold you didn't have to play because you didn't have the +buy to use it, and now we know you've skipped it. Also, you might have forgotten that Necropolis is an action.
And here's a big IRL question to add to the discussion:
How far do you let players rewind if they make a mistake?
Playing a terminal when they had another village in hand and really wanted to play two actions?
Cleaning up a duration card they played this turn?
Playing one terminal when they meant to play a different one?
Failing to call a reserve card?
I actually do allow players to rewind, but only if they have gained no new information that they didn't have before the mistake. In other words, if they play a draw card, then realize they are out of actions and try to go back and play a +actions card first, that's not acceptable, because they've looked at three new cards. However, if they play a terminal coin card, then realize they need more actions, I'm fine with them picking it up and playing the Village first instead.
Accidentally cleaning up durations: If they're drawing their cards and they're in the middle of shuffling, even if they've dealt a partial hand face-down on the table, without looking, and they're still shuffling when they realize they cleaned up their brand-new Wharf, I'll let them fish it out and put it back on the table and reshuffle. If they've looked at their cards, it's too late.
Failing to call a reserve: Even if you've played a +actions (no cards!), terminal coin and a gainer, gaining a card to the discard, then played some treasure and are wondering what to buy with $4 and shopping...if you realize then that you had a guide on the mat and you want a better hand, fine. No new information has been gained. Put the gained card back, pick up the hand, call the guide and get a new hand. Or you realize that you had a Ratcatcher on the mat and you would rather trash the copper. Fine. Pick up a copper, retroactively call the Ratcatcher and then spend $3 instead. As long as there was no drawing, no revealing, etc., there's no harm, no foul.
Some old-school card players go by the "a card laid is a card played" philosophy, but Dominion's complex enough that I feel like some grace is appropriate, so rewinding isn't unheard of in our group if it follows these guidlines.