You still have to select a climb mode
Selecting just vnav will stop it climbing at specific altitudes (as restricted in your flight plan / departure) and will resume previous climb mode once past the restriction, but you still have to select a vertical mode (FLC is the default).
So, assuming you have a “climb via the sid” and the sid is entered correctly:
Takeoff (TO/TO/TO should be annunciated). Positive rate, gear comes up, once established and clear of obstacles, engage VNAV, FLC, and HDG/NAV as appropriate. At this point the auto throttle will move to climb power, and the flight directors will start to direct you into a FLC based climb. You should see VFLC annunciated green and I believe VASEL annunciated white.
You can turn on the autopilot or hand fly at this point as you see fit, but the flight director will give you what you need.
There is some debate about FLC vs VS for climb. I’m of the very strong opinion to use FLC. In the G5000 cockpit you can tell it what speed (or let the FMS speed control apply) and it will pitch for that speed and hold it, so you are not at risk of underspeed / stall. With VS for climb, you can easily set that too high, to a vertical speed the aircraft can’t maintain. Autopilot won’t care, and will keep pitching up to try to maintain that vertical speed.
So the general rule is FLC for climb, VS / VPATH for descent.