Fractional jet First Officer reported EGPWS warnings during the climb when the flight leveled at an altitude lower than the IFR filed departure altitude due to the First Officer misunderstood the verbal clearance altitude clearance received on the ground. ATC provided vectors and issued an expedited climb and the remainder of the flight continued normally.

Narrative

In preparing to Depart PVU on Day 0; I called for clearance on the ground frequency XXX.X. The Ground / clearance controller gave us a new departure of DITTI1.DAHLI EKR climb to 5 thousand; expect 410 in 10 minutes. I needed clarification with EKR so I asked for that and read back the clearance to him and got a read back correct reply.I had further questions about the routing of the departure; so I called back for clarification and read those back correctly.I loaded the departure and set up the cockpit with 5;000 in the altitude selector. Since the airport elevation is 4;497 and the DITTI1 departure had a turning altitude of 4;997; I expected to get higher clearance on departure. We did not get a higher altitude when cleared for takeoff; so I should have asked for a higher altitude at that time. It was a clear VFR day; so we had visibility of the terrain around us.Right after takeoff at FRA (Flap Restriction Altitude) of 400 feet we leveled off at 5;000 feet. Shortly after we received audio alerts of: don't sink and landing gear. We switched to departure and checked in with our assigned 5000 altitude. They asked if we were climbing to 9;000. I replied that we were level at 5;000 as assigned. They said to expedite our climb to 9;000 feet; which we did. ATC warned us about traffic below and to our left; which we had in sight and received a traffic warning. When clear of the traffic; we received vectors to the south; inside of DITTI. Before we were switched to center we reiterated with departure we had been given 5;000 for an initial altitude. A couple of centers later; we were given an SLC TRACON number to call for a possible pilot deviation.The remainder of the flight went according to plan.The captain called the SLC TRACON upon landing and explained what happened. Suggestions: To mitigate this in the future; both pilots should listen to clearance when received by voice.