Parking is not a problem at the Belmont. There is ample on track parking, even for a crowd of over 100,000, and prices are $10-$15 for the regular lots and $15 more for preferred parking. Traffic can be a nightmare in and out of the track, but there are always lots of cops directing to keep it flowing as well as possible. You can also take the train to the track if you want to avoid driving and parking hassles. There is a LIRR (Long Island Rail Road) station right in the track at the far end of the grandstand.
Because you are in New York City, expect high hotel prices. There is only one hotel close to the track and it is always totally reserved for the horsemen, so the closest places are a good 15-20 minute drive away. Because it is a large city, there are always rooms to be had but location and price may not be ideal. Many people decide to stay in downtown Manhattan hotels and take the train in to the track. If you prefer to drive, selecting a hotel on Long Island, perhaps near one of the airports, will be your best bet. Prices per night will likely run from $100 to $150 or more.
If you want to attend one of the Triple Crown races in person, the Belmont is your best choice for both price of admission and opportunities to actually see the horses. While you will still get the best view on your television at home, you won’t be totally shut out from seeing anything like at the Derby. You will have to deal with long line-ups to bet, surly New York tellers, or the crowds that clog the track, but Belmont is a huge facility so it isn’t too horrible, other than the women’s restroom line-ups. Good luck with your wagers and enjoy the race and your chance to possibly see the next Triple Crown winner take his or her place in history.