We had friend​ vist last summer who did a solo tour. He didn't camp but stayed in various B&B's. He took the LIRR from Bayshore where he was visiting and met us in the city. We escorted him from Manhattan to Haverstraw where he spent the night in a B&B. From there he crossed the Bear Mt Bridge and stopped in Rhinebeck. Day 3 took him to Hudson NY. Day 4 into the Berkshires in Mass. Day 5 into Connecticut. Day 6 the ferry from Bridgeport to Pt Jefferson and back to Bayshore.
If you plan on camping the NYS Parks website can give you list of NY campsites. Camping - NYS Parks, Recreation & Historic Preservation
A train pass good for both Metro North and LIRR is $5