Light transport First Officer reported multiple RAs on climb out. Flight crew complied with RA advisories and continued the climb.

Narrative

Departed FRG to reposition the aircraft to ZZZ. On departure was assigned the Republic 1 departure. Tower had to queried about handing us off to Nee York Departure frequency 125.7. When we finally got on with departure as the frequency was extremely busy; we were given a point out to traffic at our 1 o'clock and 5 miles. I was able to visually acquire the aircraft. Departure then assigned us a vector that if we had turned as instructed would have put us on a direct collision course with the aircraft. We tried to make a slow turn to go behind the aircraft. New York kept giving us additional vectors that would have continued to put us on course to the other aircraft with no vertical separation. As we were about to go behind the aircraft it turned north towards us. At the same time we got an RA for an unreported aircraft that climbed up toward us from our 9 'o'clock position. While responding to that RA it drove us into an additional RA with the previously reported and seen aircraft. Never saw the aircraft the set off the initial RA but the one we were trying to maneuver to avoid initially we cut across his nose at 300 ft or less while responding to the first RA. This chain of events was likely started by the late handoff to departure and the extreme congestion around New York. Contributing to this was the multiple VFR training aircraft that chose to practice maneuvers along the FRG departure corridor. Corrective actions timely handoffs by ATC or controller prioritization in radar scope scan pattern to ensure they aren't driving aircraft towards each other; or additional assistant controller to help manage high work loads and ensure proper prioritization by scope controller.