If you want to go to Navadvipa or Puri or any other long destination in India you can take the train. There are some good trains that go from New Delhi or Nizammudim to Kolkatta or Puri everyday. For the updated schedule, fees and online booking you can check on the official Indian Railways web site. Info about the Rajdhani superfast express is available at: http://www.indianrail.gov.in/rajdhani_trn_list.html.
You can book your ticket either online or through a travel agent in Vrindavan or you can also directly go the Tourist booking office upstairs in the New Delhi Railway Station. They are open daily 8:00 AM to 5:00 PM Monday to Saturday and 8:00 AM to 2:00 PM Sunday. They close for a break from 2:00 PM to 2:15 PM. The queue (line) for paying in dollars or pounds is on the left, rupees on the right. The dollar/pound queue may be shorter. You will receive you change in rupees. If you have some small paper dollar or pound currency it will save you some time waiting for change. Note: This list was compiled from the official Indian Railways web site.
The Number 2304/2382 POORVA EXPRESS, leaves New Delhi at 4:15 PM and is scheduled to arrive in Howrah Station at 4:45 PM the following afternoon. Scheduled travel time is 24.5 hours. The train number for this train is 2304 on Wed, Thu, Sat, and Sun and number 2382 the other days of the week. This train also stops at Aligarh Junction which is closer to Mathura than Delhi station at about 6:01 PM. It is also possible to get down from this train at Barddhaman Junction which is closer to Navadvipa than Kolkata. The scheduled arrival time at Barddhaman is 2:53PM the following day. You would want to book a taxi in advance from the Navadvipa taxis to be picked up in Barddhaman. There may be taxis available there but I've never seen any.
There are local buses, but you would need to know the area quite well to use them. The SL (non-AC sleeper) is Rs. 449. The three tier AC coach is Rs. 1208.
The Number 2312 DELHI - HOWRAH KALKA MAIL, leaves Old Delhi Station (not the station nearest to Paharganj) daily at 7:30 AM and is scheduled to arrive in Howrah at 7:25AM the following morning. Scheduled travel time is 24 hours. This train also stops at Aligarh Junction at about 9:40 AM and Barddhaman Junction at 5:36AM the following day. The SL (non-AC sleeper) is Rs. 433. The three tier AC coach is Rs. 1163.
The 2302/2306 DELHI - HOWRAH RAJDHANI is the fastest and most expensive train and runs daily. It leaves New Delhi at 5:00PM and is scheduled to arrive in Howrah at 10:45AM the following morning. Scheduled travel time is 17 hours 45 minutes. The three tier AC coach is Rs. 1510.
On Fridays the departure time is 5:00PM and the train number is 2306 and that train also arrives in Howrah at 1:30 PM the following afternoon. Scheduled travel time is 20.5 hours. There are no SL class sleepers on this train, only AC coaches. This train doesn't stop at Aligarh Junction or Barddhaman Junction. The three tier AC coach is Rs. 1560.
The 2314 DELHI - SEALDAH RAJDHANI is a fast train that departs from New Delhi for Sealdah Junction station in Kolkata, rather than across the river in Howrah which might be more convenient than the Howrah trains if you intend to stay in a hotel in Kolkata. It runs everyday leaving New Delhi at 4:45 PM and is scheduled to arrive in Sealdah at 10:45 AM the following morning. Scheduled travel time is 18 hours. The three tier AC coach is Rs. 1520.
One other train worth mentioning (so you can avoid it) is the 3008 TOOFAN EXPRESS. It usually takes 36 Hours (12 hours longer than the other trains) to reach Kolkata. It leaves New Delhi at 8:05 AM, stops in Mathura at 11:10 AM and is scheduled to arrive in Bardhamman at 4:35 PM and Kolkata at 7:20 PM the following evening. The SL (non-AC sleeper) is Rs. 433. According to their website, there are no three tier AC coaches on this train. Two tier AC is Rs. 1871.