I am going to play my first round of Dominion (+ Intrigue Expansion) this weekend with 4 other players who have played Dominion before. I have read the rules and now I want to avoid the typical traps a beginner might fall into.
What are the big and tempting mistakes that you see new players make in Dominion? Is there any other advise for the first steps in this game?
The bottom line is victory points. Period. Nothing else matters. If something is not actively helping you get more or your opponents get less, don't do it. It seems obvious that I'm telling you to remember how to win, but its very easy to go super-heavy on kingdom cards until you realize its too late...
Most of the time, the way to get victory points is with $$$. When I was learning I loved to pull of the 10 card action combos that took me 14+ turns to build, but when I realized I still had only made 7$ by the end of it I started looking at things differently. Solution: Make sure you have enough powerful treasures and actions that give +coins. Gold is king!
Keep an eye on end-game. How many provinces are left? Which piles are low? What are your predictions for what people are about to buy? If you know you have the most victory points, sometimes you just want to make the game end as fast as possible. If you can use a Coppersmith to get 1 or 2 provinces early, and then use plenty of +buys cards to buy up 3 piles of cost 2 and 3 cards before your opponents decks really get rolling, go for it. Its so funny when your opponents have just built these gold stocked decks to nail provinces and they realize you just ended the game and won with 10 victory points.
Know what cards will maximize your average $ per card ratio. If you have plenty of treasures in your deck, a smithy is often even better than a gold. Likewise, if you have plenty of actions that give you +cards, make sure you have treasures to back them up.
Know the difference between early game cards, mid-game cards, and late game cards.
Watch your terminal actions. There's nothing that sucks more than drawing 5 terminal actions and having to just pick one of them to play. Try to balance your terminals with you chain actions. On the other hand, don't overload on +actions if you have very few terminals. A few of my friends love the 'village idiot' strategy: get 6 or more villages and have giant chains of them every turn. If you're not using the extra action from a village, its completely and utterly useless as all it does is replace itself.
Know the basic strategies, like drawing lots of cards, big money, attack cards, and trashing your bad cards as early as possible. These are covered very thoroughly here, so I'll just leave it at that.
Know how to evaluate a kingdom set to see what strategies will be strongest. Note how each attack card will change things, and how you need to adjust your strategy. I cover this more in the linked post.
Buy money and provinces instead of actions.
The number 1 problem I've seen beginners make in Dominion is not buying treasures and provinces.
Every time you go against the above plan, make sure you know why. There are definitely reasons not to buy a silver or gold. However, does your deck really need a possession? Or are you just buying it because it is cool?
Once you learn those simple rules (known as the "Big Money" deck) you will start to see good reasons for buying a couple additional cards to help move your deck along faster.
A second tip would be to trash copper whenever given a chance. Thinning your deck is almost always a good idea.
All the other answers are really great, so I just want to add:
Don't buy the Copper.
Copper cost 0 to buy; if you draw a dead hand (all victory points, for example), there's absolutely no shame in skipping your turn. It will ultimately end up bloating your deck and make it more likely you draw a $1 card instead of something more useful.
Also, I've seen people with multiple buys for the turn spent all their money on their first buy, and then take a free Copper.
Unless you happen to be playing with Gardens ($4 card, worth 1 VP for each 10 cards in your deck) and want to artificially bloat your deck for the chance at an extra VP, don't buy the Copper.
Biggest noob mistake is watering down your deck early.
Consider buying silver and gold as "Great" moves. Don't be tempted by all the shiny action cards.
If you get a chance, get something that allows you to discard or upgrade an existing card so you can get rid of those crappy starting cards. Don't be afraid to throw away copper! Getting rid of cheap cards is often a fantastic strategy.
If you decide to go "Pro" and take some action cards, be wary of "Dead-end" cards that don't have at least +1 action. Get a few of these in your hand and your hand will be next to useless.
Diluting your hand has been mentioned, but I would like to add its sister vice:
Not paying attention to the rate at which you can move through your deck.
or put differently
Not making sure that your next pass through your deck will only take a few turns, particularly during early and midgame play
Failing to do this can make it nearly impossible to really adjust to the different phases of the game. What those phases are depends on your strategy for the setup, but you almost always have stages you move your deck through. If you build a giant deck with multiple buys that takes 10 turns to get through, making such transitions can become almost impossible.
Failing to do this also slows the rate at which new cards actually provide additional advantage to you.
So if you are acquiring cards rapidly, make sure you are also acquiring cards that move you through your deck rapidly as well.
A lot of good points made above. In general, I'd emphasize a couple of things not yet mentioned:
Generally, in a random setup, only a few cards matter a great deal. In every game with Witch, for example, you ought to come up with a plan for playing or dealing with it. Not true for most other cards.
Generally, as a noob, you might not realize a) how important trashing is for many kinds of decks and setups...and why Chapel is so important and useful among base set cards and b) how devastating Curses are to receive for most decks.
For context, I played hundreds of two-er games back in the day, and lots of 3, 4, 5 player.
Don't be afraid to copy other players, if they know what they're doing. Don't buy coppers, don't buy coppers, don't buy estates until it's the end of the game, same for duchies in most cases.
Be sure to know the combo decks. I have played several games of dominion and dominion online, and there is nothing more fun, and more crushingly victorious, than when you drop a turn 13-ish combo that buys out the provinces in one fell swoop. It is relatively rare that these combos will come up in a kingdom set, but in order to not be a "noob," you need to be aware of them and how to beat them or they will blindside you.
A good wiki for researching these combos and anything else about dominion is wiki.dominionstrategy.com.
What are the big and tempting mistakes that you see new players make in Dominion?
A lot of great answers have already been given to this question. One thing I would add is to take several moments to really study the 10 available kingdom cards before the game starts. Look at any possible combos that you could work toward building, look for any attack cards and how they may impact your strategy, look to see if there are any trashing cards, look to see if there are any +action cards, look if there are any +cards cards, and pay close attention to any victory point cards. All of this information is very important to process before your first turn, as it will influence what strategies you should work on in pursuit of victory.
Is there any other advise for the first steps in this game?
Have fun. Winning is nice, but remember to have fun. Games are fun. Other than that, if you are up against experienced players, pay attention to what they are doing, and ask them why they are doing it. They might not tell you, but I think most people will, especially if they are your friends. As an experienced player, I want you the new player to get good so we can have more exciting games, so do not be afraid to ask questions.
