SW is much more expensive, and you will likely have trouble finding much rental selection for a house. It's mainly owner occupied. NW will likely bring you the same trouble. NE has a better rental selection. All three areas are very safe and considered desireable. A very popular rental area is more central, located north of East State Street and south of Coliseum Boulevard. This is a generally safe area and is popular for college students and families alike. It is also a very afforable rental area. I think that area is your best place to start!
SW Fort Wayne tends to be known as the rich "snobby" area. NW also tends to get this. NE Fort Wayne is more of the middle income families and is a very popular/more affordable region of the city.
Unless you are on a tight budget, I would avoid renting anything south of the Maumee river and east of Ardmore Avenue.
Hope this helps!