Air carrier pilot reported they did not climb to the minimum altitude before starting a turn resulting in crossing a ridge at a low altitude and not remaining on the charted departure.

Narrative

On the ZZZZZ1 departure from ZZZZ; we made the left turn to ZZZ [fix] at ZZZZZ instead of continuing straight to 4;200 ft. before starting the turn. The chain of events began with calling ground control for our IFR clearance. The control tower radio transmissions at this airport often briefly fades to inaudibility while the controller is speaking. As she read the clearance to us; a fadeout occurred right after she said 'ZZZZZ One'. My FO read back what he heard; ZZZZZ1; ZZZZZ2; flight plan route. The controller said readback correct. He then entered runway XX and the ZZZZZ1 departure with a discontinuity and then ZZZZZ2. I looked at it for a little while and then said that I didn't think this is right. We had a discussion about what we thought she said and we called her back to ask her what was after ZZZZZ One. She said ZZZ; ZZZZZ2. ZZZ was then entered and the discontinuity removed in the box. As a change was made and now things looked like the chart; I felt satisfied that the problem was solved and moved on. After takeoff and as we climbed through 6;000 ft. over and past the ridges north of the airport; I remarked that those hills looked kind of close today. I went back to the chart and then came to the realization that the nav had turned us at ZZZZZ towards ZZZ instead of waiting until 4;200 ft. Why did it do that? It did that because instead of going back into the departures page to add the ZZZ transition to the ZZZZZ1; the ZZZ fix was added in the flight plan; so the plane turned straight towards it at ZZZZZ; which is only there as an at or above 2500 ft. altitude crossing restriction for us. Unfortunately; another thing that added to this mistake is that the chart depicts a 001 degree course from ZZZZZ to ZZZ; so if only glanced at; gives the impression that a left turn should be made there; even though there is no segment of this departure that is an actual course from ZZZZZ to ZZZ. I pointed it out what had just happened to my FO and we had a discussion about this procedure and direct route flying in general.