I just got Intrigue, and it seems cool that it opens up the game to 5 or 6 players. However, the common consensus seems to be that its possible, but not recommended. Is there anything I can do to make Dominion play better with 5 or 6 people?
The biggest problem with extra players that I usually see is down-time. Dominion is partially a group solitaire game. There is some interaction between players with attack cards, but otherwise your opponents turns won't matter a lot besides keeping an idea on what piles are low and the game end. I find with 5 the main problem is "I know what I'm going to do, I just have to wait through 3 more people's turns to do it". You also might want to tweak the number of kingdom card piles to somewhere between 11 and 13.
The first and easiest thing you can do is lessen the cards that take longer to resolve. Spy, Throne room, Adventurer, Contraband, Counting House, Possession, and many others tend to make turns take a little longer. This doesn't mean that they are bad in any way, it just means if you are worried about long down-times with 5 or 6 people you might want to forgo them. Removing certain cards may annoy hardcore players who want a truly random layout, but my games usually aren't super serious and its worth it to fit in the extra person.
Another thing you can do if you are adventurous with 6 is play 3 teams of 2. Turns pass from one team to another. Both members of that team take their turns at the same time and have totally separate decks. Their attack cards only affect the players on the other teams. It helps to use attack cards that don't depend on order very much in case you both attack. What do you do if one of you plays a spy and the other a thief? What order do they go in? Instead of trying to make rules about this, just use attacks that don't clash with others too easily, like militia and witch. The team mechanism is mainly to allow some simultaneous turns; you can play as a team and do some minor collaboration, like who should buy what or who should focus on attacks vs. victory points, or still make it every man for themselves and have the player with the most victory points win.
Not forever, but for a while at least, don't play with the cards that require a player to make a choice for every other player. I'm thinking of cards like the Spy, where you get to choose whether or not to trash the top card of each player's deck.
I suggest this because, if your group is inexperienced at Dominion, this kind of thing can easily slow the game down to a crawl. Personally I find that a lot of my Dominion actions are really simple things like "buy a Treasure" or "buy a Province". It drives me crazy if I'm waiting 5 minutes between each move when I'm just autopiloting my strategy.
Once everyone is experienced at Dominion, of course, they can get their moves done quickly and efficiently, even if multiple additional actions and draws are involved. As long as the game is going pretty much as fast as it can go, there are no grounds can complaint. But until then, it can be really frustrating to be sitting around non-interactively for minutes on end waiting to play a move that requires no thought on your part anyway. So, yes, that's my suggestion: phase the more complex and fiddly cards in gradually!
Whenever my group has 5 or 6 for Dominion, I try to split it into two groups, either 3+2 or 3+3. With Intrigue, there are enough basic Treasure, Victory, and Curses to go around. Then when both groups are finished, you can switch and play with the other Kingdom with no additional setup time; then compare notes as to how the different groups handled the different boards.
One suggestion I have is that, once someone finishes playing all their actions, have the next person start playing. There is only one Treasure card that involves any other player doing anything, and that is Contraband, where the person to your left can name a card you cannot buy.
Having said that, if a player has a card that affects other players, they need to stop and wait for the other players to catch up. Two non-attack cards this applies to are Masquerade and Vault.
If you're going to play an action that requires you to count cards (Counting House, Philosopher's Stone), count the cards in advance. The only time you can't do this is if you have other actions that will affect the card count. In the case of Philosopher's Stone, this is any card that causes you to draw cards (cards in your hand and in play are not counted by Philosopher's Stone). For Counting House, this big one that affects this is Chancellor, as long as you have two or more actions this turn.
This next part wasn't really what you were asking for, but for each player over 4, you should add 3 Estates, 7 Coppers, 10 Curses, and 3 Provinces to the game. The Estates and Coppers make sure these piles stay at their respective starting numbers despite the number of players (12 and 32 respectively), as there are cards that may or may not manipulate these piles. Increasing the number of Curses and Provinces is mentioned in the Intrigue rulebook.