Lens for wildlife and sport photography?

I'm looking to buy i knew lens so that i can break into wildlife and sport photography. I already have 18-55mm and 55-200mm standard kit for nikon D5000 but i need more reach. I have quite a low budget as I'm 15 and this will be a birthday/christmas present. I'm wondering whether it would be better to get a fixed focal length such as 300mm f4 af-s (no vr) or a longer but not as good lens such as sigma 120-400mm. I know that the 300mm f4 is a better lens and will be good for sport such as football and track and field athletics but when it comes to wildlife photography you need a longer lens, but they are a lot more expensive. With the 1.5x crop sensor the 300mm f4 will become a 450mm, is this long enough for wildlife?
Can you please leave comments suggesting lenses that will be good for both wildlife and sport.
Thanks.

Added (1). If 450mm is long enough for both sports and wildlife then would it be better to get a 100-300mm f4 as this is the same aperture and autofocus as the nikon but it has a variable zoom which I could benefit from?

If you need to get closer to mid-range action or start photographing wildlife though, then you'll want at least 300mm at your disposal, and if you really get into wildlife, and especially bird photography, you'll want the longest lens you can afford. While the natural desire for many people is to always go for a zoom lens, don't rule-out fixed-primes. If you always find yourself zooming-into the maximum focal length, you won't miss out on any flexibility, but you'll enjoy a lens that's typically superior in overall quality.

However, the manufacturer of the lens is most important. Canon, Sigma, Tamron, Sony, and Nikon are probably the most affordable.

Wide-angle lens and high speed film are best, providing you still use an "old fashioned" camera, if not then use use a wide angle lens. F stops and such are to be used for a particular situation and will change from picture to picture.

Wildlife and sports photography are two different animals…

but you can use the same kind of lenses…

Sigma 150-500mm is good for close work from a good distance…