According to the rulebook this phase is resolved simultaneously:
Once every player has selected his or her card, they perform their action
simultaneously. Three actions are possible with the chosen card :
- a. Build the structure
- b. Build a stage of their Wonder
- c. Discard the card to gain 3 coins
In other words, all players announce their action, build/wonder/discard, at the same time. It's out of the spirit of the game to see what other players are doing and then change your mind. For example, you might have two red cards at the end of Age III and you'd really like to know if your neighbor is going to build one, so that you can discard yours for money or not. You have to decide before phase #2.
When most players are newbies, it's a commonplace to resolve the cards one at a time, so that they can follow, but you should normally resolve the actions simultaneously.
Technically, it's also illegal to decide which neighbor to pay after resolving the actions. You're supposed to decide beforehand who to pay; for example, you cannot decide to pay Olympia, see that he just built the Arena, taking $9, and then decide to pay your other neighbor.