Others have covered transportation from the city. If you are coming via car, I recommend parking in the "Green" lot off the Cross Island, and taking the shuttle bus into the track. If you stay for any of the races after the Belmont, it's not too bad getting back out to the lot via the shuttle buses, and the police have done a good job limiting traffic in the inside lanes of the Northbound Cross Island Parkway after the race, making it far easier to get out than if you park in a lot that forces you to get onto Hempstead Turnpike.
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