Which is the right lens for me

My birthday is soon and I'm looking to get a new lens. I have a NIKON D5000 and 18-55mm and 55-200mm kit lenses. I have a choice of three lenses, NIKON 300mm f4 d af-s if-ed, Sigma 100-300mm f4 af-s or Sigma 120-400mm f4.5-5.6.
I'm breaking into sports and wildlife and so I need quite a long lens. The Nikon d5000 has a 1.5x crop factor so all the lenses become longer.
Which of these lenses best suits my situation?

Wildlife and sport are different photographic requirements…

it will depend on:
1) the wildlife you are shooting and where
2) the sport(s) you are shooting and from where

To be honest we can't really answer your question, wildlife and sports are wide subjects, some wildlife require a Macro lens for instance.

For indoor sports a wide aperture lens is useful in the low light as the lens will auto focus quicker, my favourite lens for this is a Sigma 70 - 200mm f2.8, that same wide aperture also makes it one of my favourite portrait lenses as I can selectively focus on my subject at the wide aperture and throw the background out of focus, but it hasn't got the reach required to photograph say a small bird 60ft away for that a 500mm lens (and a good tripod) may be needed.

The Nikon f4 being a prime lens, should give you better image quality, the Sigma 100 - 300mm gives you more options in focal length, but is beginning to sacrifice some aperture, the Sigma 120 - 400mm has even further reach but is beginning to be a 'bright day' lens, indoors it will have limited usefulness as the smaller aperture limits it's ability to auto focus quickly (if at all).

All lenses focal lengths are basically compromises. No one lens will do everything that you want with either subject. By and large always go with the wider aperture lenses, but wide apertures can compromise image quality too, at least the lens has a better chance of achieving auto focus quicker in low light, it has more chance of getting the shot you want in focus, and you can always use a smaller aperture to get the image quality back at the expense of a longer shutter speed, it's all swings and roundabouts, there's no one simple answer.

The Sigma 100-300 mm f/4 would be an excellent lens for shooting sports, but do you know that the lens costs $1, 800?

The Nikkor AF-S 300 mm f/4 IF ED is a great lens too and costs less than $1, 350. I use the 300 mm f/2.8 for most of my night and indoor games.

Sample

The 120-400 lens is just too slow for shooting sports at night or indoors.

Even the two f/4 lenses are a little slow, so you will have to learn to shoot at the peak of action.

Here is an other shot, shot indoors using the 300 mm f/2.8

As you can see, at f/2.8, the shutter speeds are 1/500th and 1/640th second. Using a f/4 lens, those shutter speeds will be twice as long, 1/250th and 1/320th second

Both the f/4 lenses are fine for shooting wildlife. You will need a good sturdy tripod and a good blind that is set downwind of the gathering places of the wildlife.

Once you buy your lens, it could take a few games or sessions shooting wildlife before you will master using these lenses

Skip over the Sigma 120-400 but the other 2 are fairly interesting for your fieldMost people will recommend the Nikon 300 f4 but I like the flexibility of a zoom so I. Recommend the Sigma 100-300 f4… But first go see if you can find some decent reviews on sites like DPReview. It's quite possible that the 100-300 has absolutely horrible image quality, you can;t just judge a lens by its stats.

There's no contest. Nikon lenses are bye far the better bet. Remember CHEAP glass usually means degraded images. For sports photography your best bet is a long focal length zoom lens and as fast as possible.e. G 70-300mm and for wildlife go for a long focal length macro zoom which will keep you physically away from the subject and still give you macro shots without the possibility of you casting shadows on your subject.

In the good old days one could leave a substantial deposit with a retailer and test the lenses before purchasing. I have yet to see any info on lens performance these days. It is still possible to do your own lens test with the use of lens test charts.