There are two things you need to do right after takeoff, just as a matter of getting the airplane converted from ground vehicle to air machine, and there is one thing you need to do just as a matter of keeping safe. Everything else can wait. How long can it wait? Our answer is 10,000 feet but there are other views.
There is a bit of heresy here for those of you with deep Gulfstream roots. I have been flying one shape or form of Gulfstream since 1992 and in all that time have begun this checklist with gear up, ground spoilers off. Twenty years later I've reversed those steps. See "step 0" below for why.
Everything here is from the references shown below, with a few comments in blue.
[G450 Airplane Flight Manual §2-05-10]
There are generations of Gulfstream pilots who rightfully connect the ground spoilers switch with the gear handle, knowing that bringing the power levers towards idle with a Weight on Wheels System Failure can be fatal. Like them, I used to always select gear up and then ground spoilers off. But what happens if you do have a Landing Gear Failure to Retract and the gear handle doesn't come up? Won't you be confused enough to forget the ground spoiler switch? What's the next thing to happen if the gear is stuck down? Chances are the guy in the left seat is going to be leveling off pretty soon to bring the airplane around for a landing. When he does that the autothrottles are going to bring the power levers back. If the WOW is in the ground mode, those spoilers are coming up, aren't they?
I usually wait for the altimeter to start climbing, the VVI to head north, and a visual confirmation that we are heading up. That usually takes a second or two. If you didn't do so in step 0, immediately turn the ground spoilers off. Sure, there are several systems designed to keep the spoilers stowed when airborne, but connecting the spoilers to the gear handle in your procedures only takes a second and it can save your life if other things go wrong.
Limit bank angle to 15 degrees while retracting flaps until reaching V2+20.
[14 CFR 25 §25.111(4)] The airplane configuration may not be changed, except for gear retraction and automatic propeller feathering, and no change in power or thrust that requires action by the pilot may be made until the airplane is 400 feet above the takeoff surface.
[G450 Aircraft Operating Manual §06-02-40] Flap retraction: "at 400 ft abv airport level (AAL)" but it also notes: "Consideration should be given to delay flap retraction during heavy weight takeoffs/turns/terrain avoidance in order to retain takeoff power setting."
Some pilots argue that 14 CFR 25 is for aircraft manufacturers and they don't have to follow the 400' rule. I wonder why they are flying their airplanes outside of the design criteria. Some flight departments use 1,500 feet for all takeoffs. We use 400 feet, as the AOM says, unless there are other considerations, such as a Noise Abatement Procedure or obstacle clearance considerations.
This step should be item 2 at the latest, item 0 for me.
Figure: Cabin pressure indicator panel, from G450 Aircraft Operating Manual, §2A-21-00, figure 10.
In cabin pressure AUTO you should see the cabin altitude start and the differential pressure increase. In MANUAL you should at least see the differential rise. Don't bother with the synoptic, there is a gauge right above your head.
We do this at 10,000 feet.
14 CFR 25, Title 14: Aeronautics and Space, Airworthiness Standards: Transport Category Airplanes, Federal Aviation Administration, Department of Transportation
FSI G450 PTM, FlightSafety International Gulfstream G450 Pilot Training Manual, Volume 2, Aircraft Systems, October 2008
Gulfstream G450 Aircraft Operating Manual, Revision 35, April 30, 2013.
Gulfstream G450 Airplane Flight Manual, Revision 35, April 18, 2013