Go greyhound from South Station-
or Amtrak Acela---will get you there in a few hours
(from Lowell you could take the Commuter rail to North Station - Orange line to downtown crossing or Green Line to Park St and change for the Red Line (Ashmont or Braintree Train) to South Station or if you didnt want to take the subway you could get a cab from North Station to South Station from outside the north station there's cabs by the garden parking lot across from the 7 eleven on Causeway street out front.
Might be quicker to go to North Station commuter rail from Lowell
to North Station and get a cab to South Station
then get the Acela Train to New York
---
if you really needed to save money you could do greyhound or peter pan bus from South station
commuter rail lowell line
to north station
cab to south station from north station
and then greyhound bus or peter pan bus to NYC
will be a couple hours longer than the Amtrak Acela train but less expensive
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Hope this helps
Here's a list of top cheap hotels in NYC
Recommended Cheap Hotels in New York City - Guide to Cheap New York City Hotels
Hope this helps