Kolkata is the capital of West Bengal. For the Navadvipa Dham Parikrama, this is the closest place to fly into. Kolkata definitely has a vibe of it's own. 'Frenetic' might begin to describe it. It's like the whole city should switch to decaf.

From Kolkata airport, train station or Burdwan Junction you can book a taxi to Navadvipa for  about 1300-2900 rupees, depending on which car you prefer and your orign. All types of vehicles, from cars up to buses including AC Sumo or Scorpio and Non-AC Sumo and Scorpio. This is the best option by far for most devotees.

In the morning there are four 'good' direct trains from Nabadwip to Howrah: 4:18 AM, 6:14 AM, 6:53 AM and 7:31 AM. The first two are a bit faster, reach Howrah in less than four hours.
Note: Tickets for these trains must be purchased in advance at any computerised booking office.

Sri Gour Purnima is the same day as Holi, a festival where mad revelers throw permanently staining DYE at everyone. This day is therefore not a particularly good day to travel. In Barddhaman (Burdwan) the festival of Holi is celebrated one day later. There is a computerised booking office at Nabadwip Railway Station. It is open 8:00 AM to noon, and 12:30 to 2:00 PM. except Sundays. Tourist quota tickets are only available in Kolkata and New Delhi.

From the Howrah train station (platform 1 to 6) you can take a train to Navadvipa.

The best train to Nabadwip is the Train Number 2065 JANASATABDI EXPRESS leaving Howrah at 3.25 PM daily except Sundays and Holidays. It is scheduled to arrive in Nabadwip at 5:17 PM. There are two classes of service on this train, Second Seating (Rs 67) and AC Chair Car (Rs 215).

There are two classes of trains available. One is an express train, and the other is a local train. The express train, besides being faster, is more comfortable and better if you are carrying luggage. There is also a lower chance of theft on these trains. Howrah station and the local trains are quite notorious for pickpockets and luggage theft, so be very careful with your valuables!
There are some good trains that go from Kolkata to New Delhi everyday. Fares quoted are single (one-way) fares from Kolkata to New Delhi. Note: This list was compiled from the official Indian Railways web site.

There are loads of internet places in the tourist areas. Netfreaks, 2/1, Sudder Street, phone 246-9586. Cyber, 61A Park Street, phone 217-3645. Cybercafe, 13, Chowringhee Lane (off Sudder St), phone 245-8675 Junction 96 has a number of locations including 20 Park Street, phone 216-0331

As mentioned before, most Bengalis aren't strict vegetarians. Pretty much all of the pure Veg places are run by South Indians (who mainly are strict vegetarians). There are other places mentioned in the popular travel guides, but these places are purported to be pure-veg and are at least somewhat close to the hotels (above).

If you have arrived at an odd hour there are many hotels in Kolkata. You can pre-book a hotel room by phone before you come to India. The phone numbers of decent hotels are in the India travel guides some of which can be found on the Travel Planning page.

NOTE: In Bengali "Rickshaw" is pronounced "Rish-kaw". Try not to laugh too much when the time comes.

Yes, they exist. They may be very crowded and uncomfortable, and there is no provision for your luggage. It costs about Rs 10. It would be better to take the taxi.
This is not a recommended option and having anything more than hand luggage would be a problem. There are buses from Kolkata to Krishnanagar. From there you would need to take a different bus to "Navadvipa Rail Gate". From there you would need to take a rickshaw. Buses depart from the airport "Ek Nambar Gate", an Rs 10 rickshaw ride from the terminal. The buses are normally going to Malda or Borampur with a stop in Krishnanagar.