Dress well and be prepared for any questions the landlord may ask. If you are not confident speaking Spanish, try to bring along someone to help out. As a foreigner, you sometimes need to be extra convincing.
You might find it useful to go to appointments with cash in hand, particularly if the apartment was advertised in a newspaper like Segundamano. If you take a day to decide, you might well find your dream flat already taken by someone else. Don’t let the competition scare you into signing a contract unless you are sure, but when a good flat comes up, you need to act quickly.
If possible, try to visit the area around the apartment both during the day and at night, or ask around to see what it is like. Perfectly peaceful areas during the day can turn into unbearable residential areas at night if there are bars nearby. Likewise, a calm neighborhood in the evening may be a nightmare during the day due to traffic or construction works.