I had the 70-300mm non-VR lens, and it was poor. A very cheap build, and very soft at 300mm. I took sample pictures on a tripod at 300mm, and compared them to my 80-200mm at 200mm, and decided it was almost better to shoot with the 200mm and crop (in other words, cropping the 200mm shot so the subject was the same size as the 300mm shot, the 200mm was about the same sharpness).
The lens autofocuses very slowly due to the minimum aperture, especially at 300mm (f/5.6).
The Nikon 80-200mm f/2.8 is a good choice if you want to try a used lens, as they are built like tanks. Could add a 1.6x teleconverter. The 80-200mm is a good lens for sports (fast AF) and portraits (nice bokeh) too. There are AF-D and AF-S versions, you'll need the AF-s on your D40.
The Sigma 150-500mm is a good lens. It's also slow, but has OS (VR) and a lot of reach. You can hand hold it at 500mm in good light. It is an HSM lens so will AF on your D40.
The Nikon 70-300mm VR is a great lens. It's completely different than its non-VR brother.
It's AF-S so will AF with your D40.