1999 Chevy Suburban

Do you want to remove the pump, or the gearbox?

It's a little different for the different engines.

What size V8 do you have?

And do you have a C1500 or 2500?

DAMN. 99 was the cross over year.
They switch from 5.7 to 5.3 mid year. So i'm going to give you instructions for 1998 5.7 and 1999 5.3.

Be back in a few minutes.

1998 5.7
Remove upper radiator fan shroud.
Unload power steering pump belt tensioner and remove belt.
Place drain pan under steering pump.
Disconnect return and feed lines from pump
Cap end of lines and pump fittings.
Remove bracket mouthing nuts.
Remove bracket.
Remove pulley from pump shaft, (you will need a PS pulley remover that you can rent at most auto parts stores if you don't have one, most of the time it's basically free, you leave them a deposit on the tool and then get your money back when you return it.)
Remove pump mounting bolts
Remove pump

Install in reverse procedure.
Use pulley installer to install pulley, (same deal with renting one from the parts store)
Install pulley flush with end of pump shaft +/- 0.010"

1999 5.3
Remove upper radiator fan shroud.
Unload power steering pump belt tensioner and remove belt.
Remove pulley from pump shaft, (you'll need the rental tool again)
Place drain pan under steering pump.
Remove the hose guide bracket from the front of the pump.
(for vehicles with variable effort steering, disconnect the electrical connector.)
Remove hoses from pump.
Remove the lower intermediate shaft from the steering gear.
Remove the bolt retaining the rear bracket to the engine.
Remove the bolts from the front of pump
Remove the pump from vehicle.
Remove bracket from the rear of pump

Install in reverse order.

They both have a very convoluted bleeding procedure. And I will type it all out if you want, but that's not how I do it.

I jack the front end up, so the wheels are not on the ground.
Start truck and turns wheels very slowly, back and forth from lock to lock about 10 times. Just barely tap the lock each way, DO NOT hold it there.

Let me know if you want the long version.