And so, in summary as I just experienced this exact problem with a Leviton 20 Amp GFCI Outlet that was the primary outlet and protected 3 other downstream outlets.
If you follow the Leviton directions and have 4 or 6 wires coming into the box (NOT counting any grounding bare copper wires) you MUST make sure the LINE and LOAD wiring is connected to the right terminals.
IF after you wire this outlet and the solid green light comes on but there is no power to the GVCI outlets nor is there power to the connected downstream outlets, nor with the reset/trip buttons work, YOU HAVE NOT CONNECTED THE LINE WIRES TO THE LINE TERMINALS AND THE LOAD WIRES TO THE LOAD TERMINALS. The solid green light merely means the outlet has detected power somewhere in it's circuits. RED or FLASHING RED light indicates a further wire connection problem.
To correct the situation above, after plugging a lamp or light into the GFCI (in tripped position) turn the appropriate breakers off at the circuit box, then reconfirm you know which are the LINE Input Wires and connect those to the appropriate LINE terminals. Make sure copper ground is connected. THEN (keeping any other wires separated) turn the circuit back on at the circuit box. You should see the solid green light and when you reset the tripped button by pushing the black button in, your plugged in light should go on.) Turn circuit off again at circuit box; reconnect the LOAD wires reinstalling the outlet and cover plate in the wall. Turn circuit on again and you should get the green light, power to the GFCI outlets AND power to any downstream outlets that get their power and are protected by the GFCI outlet.