B737-700
Flight Crew Multiengine; Flight Crew Instrument; Flight Crew Air Transport Pilot (ATP)
Flight Crew Last 90 Days 300; Flight Crew Type 15000
Situational Awareness
Aircraft Equipment Problem Less Severe; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter Unstabilized Approach; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Provided Assistance; Flight Crew Became Reoriented; Flight Crew Executed Go Around / Missed Approach; General Flight Cancelled / Delayed
Aircraft; Software and Automation; Environment - Non Weather Related; Human Factors
I was the Pilot Flying on the ZZZZZ 1 Arrival into ZZZ. ZZZ TRACON asked us what approach we wanted to [Runway] XXR into ZZZ. We requested the RNAV visual to [Runway] XXR. We were cleared to 7;000 feet and cleared to intercept the final approach course for [Runway] XXR after ZZZZZ. We thought we were cleared for the approach so we set the next altitude of 4;000 feet in past ZZZZZ.  After crossing ZZZZZ the aircraft started descending to 4;000 feet. ZZZ TRACON informed us of a Low Altitude Alert and had us climb to 7;000 feet; then he Immediately cleared us for the visual. Told us to take down the phone number for possible pilot deviation.
Both our first time to ZZZ.  We were cleared to intercept final segment. We said to each other; 'ZZZZZ is IAF;' and then we determined it was the final course we were supposed to intercept.  We should have asked ATC to repeat; but we put 4;000 feet in the autopilot; as in VNAV; to continue published approach; but 7;000 feet was still our clearance limit.  On the published segment it continued down; inside ZZZZZ; below 7;000 feet.  ATC queried us at 6;000 feet and told us to climb.  We had the field in sight and ATC then cleared us immediately for the visual.I was so focused on terrain and being in VNAV. As Pilot Monitoring; I should have questioned the clearance and any confusion we had. I am sure what we heard; thought we heard and what ATC said were all different.
B737-700 flight cew reported a low altitude alert and initiated a Go Around at an airport that both crew members were flying into for the first time.