Any Unknown or Unlisted Aircraft Manufacturer
Flight Crew Instrument; Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP)
Time Pressure; Workload; Communication Breakdown; Confusion; Situational Awareness
Conflict NMAC; Deviation - Altitude Excursion From Assigned Altitude; Deviation - Altitude Overshoot
Air Traffic Control Issued New Clearance; Flight Crew Returned To Clearance; Flight Crew Took Evasive Action; Flight Crew FLC complied w / Automation / Advisory; Flight Crew Requested ATC Assistance / Clarification
Human Factors; Procedure; Airspace Structure
On departure from APA [we were] initially given runway heading. Soon after departure; tower gave [us] a clearance to turn and climb to the east along with a traffic advisory of traffic entering the downwind. TCAS showed a traffic conflict with traffic within 3 miles and 300 ft. We requested a runway heading until clear of traffic. Afterwards; tower cleared to turn east and contact TRACON. TRACON gave [us] a traffic alert with traffic within 700 ft. of our assigned Altitude. Crew requested a climb to avoid traffic converging traffic. Traffic then advised as maneuvering and shown on a TCAS within 400 ft. Last call of traffic 400 ft. below us and gave us direct fix ZZZZZ toward the previous reported trafficking. Many aircraft within airspace while multiple clearances were blocked to include our clearance to ZZZZZ. A clearance to maintain 9;000 ft. was received. At some point both crew members agreed 10;000 ft. was the altitude. A short time later we received an RA and ATC gave an immediate descent and turn to heading 360 and maintain 9;000 ft. ATC gave a phone number to call once on ground. Two close encounters within 4 minutes of takeoff with VFR traffic: one on takeoff and one before turn to ZZZZZ; both less than 500 ft. separation.Additional TRACON frequencies [are needed] during high workload times. Multiple blocked transmissions and VFR flight following request in short time on this frequency. Crew both agreed 10;000 ft. selected in the window. Potential wrong altitude set and agreed upon in altitude selector.
Pilot reported multiple traffic conflicts shortly after departure.