Is it possible to go from Philadelphia PA to Purchase NY using any
kind of public transporation like buses or trains or subways also
using a cab for as short a distance as possible? Thanks

You begin by getting (in Philadelphia) to 30th Street Station.
http://www.30thstreetstation.com/ ; taking a train to NY Grand Central
Station. You can check fares with Amtrak in this site:
http://www.amtrak.com/ - there are different fares for students,
children, seniors, etc. - so you might want to check out if you
deserve a discount.

For example, I checked for today, and found 10 trains to New York. 
http://tickets.amtrak.com/JBookIt/?function=handlers.amtrak.AmtrakTrainAvailabilityHandler&storefront=1003&trueAvailabilityRequest=yes&inputAmtrakSavedRequest=handlers.amtrak.AmtrakMainSimple&inputDepartureDate=Jul+9%2C+2002&inputReturnDate=&pickedCityIndex=&inputDepartureStation=Philadelphia%2C+PA+&inputDestinationStation=NYC+Grand+Central+Terminal&departMonth1=Jul&departDay1=9&inputDepartureTime
0700&returnMonth1=&returnDay1=&inputReturnTime=&inputNumberOfTravelers=1

In New York, you take The Metro North railroad from NYC Grand Central
Terminal - take the Blue line ("Harlem" line; end-station is Wassic).
You get down in White Plains. See more info here - www.mta.nyc.ny.us/

In White Plains, you take a bus "Anderson Hill Road bus No. 12" (or
take a taxi).

I hope that helped, please contact me if you need anything else. I
also leave you with my search terms:
Purchase ny 
Philadelphia PA train nyc 
getting from Philadelphia PA  nyc 

And a couple of sites I found the most helpful: 
http://www.mville.edu/about/directions.html#6 
http://www.purchase.edu/admissions/adm_vis_directions.asp

I may be wrong, but it is my understanding that Amtrak trains run to
NYC Penn Station, not Grand Central.

Penn Station is at W. 34th St and approx. 6th Ave., while GCT is at
42nd St. and 5th Ave. It's not a bad walk from one to the other, or
you can take the Metro or a taxi. To take the Metro, you take the
1/2/3/9 (red) line North from Penn Station one stop to Times Square,
then take the S (shuttle) or the 7 (purple) from Times Square to Grand
Central.

--Joanna

Much cheaper (but a little longer) to take the R7 from 30th to Trenton
then NJ Transit to MSG.

The Purchase College website has a "Directions" page with travel information at:

http://www.purchase.edu/admissions/adm_vis_directions.asp


Worth a look.

I would take the Greyhound bus from Philadelphia to White Plains, NY.
That costs $26 one way, and the trip time is between 4 and 5 hours
depending on the time of day. Buses depart at 1:35am, 7am, 9am, 1pm,
4pm, and 11:30pm. (The 11:30pm bus takes much longer than the others,
with 6 hours and 15 minutes of scheduled travel time.) Online purchase
may or may not be available, depending on how soon you plan to travel.
You can also buy the ticket in advance over the phone, and pick it up
on your day of departure. This would save you a trip to the station as
well as waiting in the longer ticket line. For all the details and
schedules, go to:

http://www.greyhound.com

Once you get to White Plains, according to the site davidsar-ga
provided: "Take the No. 12 bus Northbound from White Plains or
Southbound from Yorktown, Mount Kisco, or Armonk. Buses leave
approximately ever hour, Monday through Saturday."