Be prepared to rent the apartment on the spot. In the current market, rental apartments rarely stay available for more than a couple of days. If you are not prepared to rent the apartment, do not bother visiting it.

Bring the following: letter(s) from previous landlord; pay stubs; bank statements; and a check book.
Be prepared to put down a deposit of one month's rent.
Check the neighborhood around the building. Look for hospitals, fire or police stations - they are noisy and will cause you to lose sleep.

Check the space, layout, and quality of the apartment. Make sure you feel comfortable.

Ask any other questions that are on your mind.