On the NG LNAV and VNAV can be armed on the ground. Some airlines arm only LNAV on the ground and it will engage at 50 ft, others engage LNAV at 400 ft AAL. VNAV is engaged when Flaps are UP.
For take off, pitch initially to 15 degrees then follow Flight Director guidance.
As for rates of climb, I have never really paid particular attention. We climb at a fixed speed which changes depending on the Cost Index and wind inserted in the INIT PERF page in the CDU. Standard is 250 up to FL100, 280/.78 above.
TOD FLx3+10 works good enough for average weights with a standard descent profile of .78/280/250 to cross check VNAV is behaving.