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.