Air taxi Captain reported a near miss with another aircraft during climb under ATC vectors. The PF followed the TCAS RA and ATC vectors to avoid the other aircraft; then rejoined the standard departure.

Narrative

The Napa (APC) tower controller cleared us for takeoff on runway 24 and also amended our initial cleared altitude to 5000 feet. Shortly after takeoff we were handed off to OAK center; as we were climbing on the Ozzie4 departure procedure as cleared. The Oakland Center controller said Radar Contact" and advised us of VFR traffic at our 2 O'Clock position. I immediately asked the controller "If you need to turn us for traffic separation; that is ok". He immediately amended our altitude to 3000 feet and told us to turn right to a heading of 160 degrees. I queried the RIGHT turn in the pilot readback by repeating "Right Turn" twice; as it appeared that a right turn was towards the converging VFR traffic observed on TCAS and was also the long way around. The controller quickly corrected himself and said turn "LEFT" heading 160. We immediately received a TCAS RA to Descend. The Pilot Flying started his descent in his left turn to 160 degrees. The Controller was immediately notified were descending for a RA advisory and he then cleared us down to 2000 feet. We never saw the converging VFR traffic; but it appeared to cross above us at approximately 400-500 feet on the aircraft TCAS/ MFD (Multi-function Flight Display). When clear of the VFR traffic (and still in our descent to 2000 ft and left turn to 160 degrees); The ATC controller gave us a new clearance with a new heading and altitude to climb to; as we started a new vector to rejoin the Ozzie4 departure.Airplane TCAS RA alert. Cause: VFR converging traffic near our cleared departure route; shortly after takeoff.Pilot responded to a "Descend" TCAS RA.Suggestions: When possible; have ATC (Oakland Center and/or NorCal Approach) provide better IFR separation from VFR traffic; for departing IFR traffic from the APC airport."