If you jump on an InterCity (IC) train from Amsterdam (€17.20 one way) you will get to Dordrecht in under 90 minutes. You can easily purchase tickets from www.ns.nl
Besides a nice ramble around the city centre 'binnenstad' to see one of the hundreds of monuments, at this time of year it is also nice to drop in to the Dordrechts Museum (and on a side note I do recommend the museum card if you plan to visit more than 2 museums during your holiday).
There are so many gems in this museum but my favourite one (perhaps due to the time of year) is a painting from the 17th Century by Dutch Painter Albert Cuyp. You probably recognise the name if you have spent anytime in the Netherlands. The painting is called 'Vrouw met schaal oliebollen' or woman with pot of oliebollen.
This painting is from the 1600's so eating oliebollen to usher in the new year is not only a nice and tasty tradition, but one that really makes you feel you are living in continuity with past Dutch generations!
Don't forget to check out the VVV Dordrecht website for info on what to do when in Dordrecht.
So Happy New Year I hope you get to visit Dorcrecht or at least have some Oliebollen to usher in the New Year!