Which nikon lens is best?

I'm on a limited budget and would like to enter the world of sports photography and hopefully eventually get into wildlife and would like to know which is the best lens for this that is under $600.
I'm not sure if this is possible but if you know of any then please tell me…

AF-S Nikkor 70-300mm f4.5-5.6G VR IF-ED is the only lens worth considering in your price bracket. It's a well made, well designed lens, with a quck to focus IF design. Tamron makes a similar IF construction lens (The SP 70-300mm f4.5-5.6 Di VC LD USD) which is also worth a look, but Nikkors will give you better compatibility in the future.

The reason IF is important is that inner focusing (and rear focusing) lenses are far faster to focus than traditional designs. IF means you only move the focus group, rather than the entire lens or the entire front group, which means the bit that moves is lighter and needs to move less distance therefore AF becomes much faster.

The 70-300 VR comes to mind, not ideal but a very good consumer zoom.

If you really hunt? You could grab a Nikkor 400mm f/3.5 but it's a manual focus so that's a sad drawback for some. For even less a Nikkor 400mm F/5.6 that's manual and slower. Sacrifice is your key consideration. What are you willing to live without? If you must have "IQ" as your #1 priority then manual focus Prime Lenses.AF? That makes it difficult. I would recommend the Nikkor 300mm F/4 older versions. You could use a 1.4x or 1.7x teleconverter on one and still obtain Pro quality.

No way would I recommend the 70-300mm VR for Pro IQ. The 80-400mm is a little more above your price level but it does grab high level IQ well above the 70-300mm VR to make the extra cost well worth it.

Stay low if you go 3rd party. The zooms simply do not deliver the depth, contrast, and clarity. Sharpness is very good on some at F/8 and mostly F/11 but depth, color, contrast? Blah! So F/2.8 primes are superior in the 3rd party realm… Tamron 300mm F/2.8? AF or Manual Focus is a winner. Avoid F/5.6 3rd party primes or kiss IQ goodbye. 50-500mm, 150-500mm, 170-500mm, 200-500mm, 400mm? Sigma, Tamron, or Tokina? 1 in 50 chances you'll get Pro IQ in a shot, tried em all. The 200-500mm is the best. The 400mm Sigma APO Macro is hard to find but raises your chances to 1 in 20 maybe. Bigma's? Please… They are too risky for getting Pro IQ…

If I was still teaching? I'd still teach people to use Manual Focus and then move up. Here's why:
you will become extremely more aware of everything inside your viewfinder, although without a split focus screen the little green dot or moving points is annoying. You will learn patience and become more selective. Or just use a fast focusing lens and high speed camera and don't even look through the viewfinder, just point it in some direction and hold down the shutter like a Wildlife/Sports Paparazzi Photographer? (why they just don't use automated robot tripod fired stations for today is unknown, but that's what these guys are today… Shutter button holder downers)

Read this, it's subjective but it is very trustworthy - http://www.naturfotograf.com/lens_surv.html

If Bjørn Rørslett rates it a 4 or better (make sure you apply the rating to what you use as close as you can, DX, FX, Film) I can assure you it will deliver Pro IQ…

If you would take the Manual Focus Nikkor F/5.6 and blow up shots to 100% or even 200% you would clearly see just how poor most consumer end 3rd party and even Nikkor lenses are.

For a really clear indicator of long glass? Simply view the market and pay attention to lenses that are for sale - the more? The less desired. Speaks volumes, the market does…

Nikon 70-300mm VR lens is a good choice for your budget. Good zoom length, Fast Auto-Focus when working properly.

You should go with Nikon 105MM AF-S VR II F2.8D Micro Nikkor Lens

Product Description: Nikon Micro-Nikkor macro lens - 105 mm
Lens System: Macro lens
Intended For: 35mm SLR, digital SLR
Mounting Type: Nikon F
Lens Aperture: F/2.8
Focal Length: 105 mm
Min Focus Range: 31 cm