B737-700 First Officer reported the flight took off with the pneumatic bleed valves closed; resulting in failure to pressurize. The crew noticed and corrected the error.

Narrative

We turned the engine bleeds off while taxiing into the gate from our first flight; to prevent a dual bleed light. The First Officer forgot to turn them back on after engine starting. We performed the appropriate checklists and the appropriate calls were made; but we forgot to visually check the bleed sources were in the correct position; when the Captain responded to the checklist. We climbed out as normal; but as we were performing the 10;000 ft. Checklist; the Cabin Pressure Warning horn and light went off. We leveled off at 10;000 ft.; told ATC and immediately realized what we had done. We turned the bleeds on; went through the QRH to resolved the issue; and the cabin pressurized as normal. We notified ATC and the Flight Attendants; continued the climb and proceeded on course as intended. No others issues entailed during the flight. Threats to this situation were an extremely early report time and early morning awakening for both Crew Members. First Officer was new and Captain visually looked at bleeds during checklist and responded with what was 'expected' but not as the bleeds actually were.Making sure that you respond to what you see not what you expect during checklist callouts And verifying bleed sources are in the correct position; after engine starting and shutting down the APU.