Your conversation is perfectly fine.
"Do you want to go to the Italian restaurant?"
This implies that there is only one Italian restaurant that the speaker could be referring to. The speaker is assuming that the listener knows that there is only one and does not need further clarification. Often, sentences like this are followed by qualifying prepositional phrases like so that the listener can determine which specific restaurant the speaker is referring to:
Do you want to go to the Italian restaurant in the mall." OR
Do you want to go to the Italian restaurant down the street.
"Do you want to go to an Italian restaurant?"
This implies that the speaker does not have a specific restaurant in mind, regardless of how many there actually may be (in fact, this sentence is acceptable to use even if there are no Italian restaurants).
So in summary, if there is one restaurant and both the speaker and listener know this, you should use Do you want to go to the Italian restaurant? In all other circumstances, you should use Do you want to go to an Italian restaurant?