Commercial Fixed Wing
Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Conflict Airborne Conflict; Deviation - Altitude Excursion From Assigned Altitude
Air Traffic Control Issued Advisory / Alert; Air Traffic Control Issued New Clearance; Flight Crew FLC complied w / Automation / Advisory; Flight Crew Took Evasive Action
Airspace Structure; Environment - Non Weather Related
We were being vectored for the ILS to runway 4R in Newark.  We were southwest of the airport; level at 3000'.  ATC gave us a traffic advisory for a VFR aircraft 500' below our altitude.  No more than 5 seconds after that advisory; we were given a TA from the TCAS; followed immediately by an RA to 'Climb'.  I disengaged the Autopilot and began to follow the RA guidance while the FO let ATC know we were in a TCAS climb situation.  ATC acknowledged and told us to climb no higher than 3500'; as there was other traffic above us.  As we leveled at 3500'  we received another RA to 'monitor vertical speed'.  This command was for us to just stay level at 3500.  I then saw the VFR traffic (appeared to be a Cessna model) pass directly below us at what looked to be less than 1000' lower than our current altitude.  After clearing the traffic; ATC issued us a clearance back down to 3000 and we continued towards the approach.The main threat in this situation is congested airspace; with the addition of light GA traffic on days where the weather is nice.  I am unaware if an error on the part of the VFR pilot contributed to our TCAS to generate an RA.I will continue to maintain a vigilant scan for traffic; especially in good VFR flying days.
First Officer reported maneuvering for a TCAS RA; while being vectored for an ILS to EWR; that resulted in a second RA with another aircraft.