As someone who learned to drive on an '83 Suburban, I feel well-qualified to answer this question! Between my parents and myself, we've owned 3 Suburbans (an '83, '87, and '95) and seven half-ton Chevy trucks, so I have a great deal of experience with these vehicles.
I have a Hotchkis TVS system (F&R sways) on my '94 Chevy K1500 (1/2 ton, 4wd, regular cab short bed). It has far less body roll than my '97 328is with the factory sport package
I'd make sure your steering components and bushings are in good shape first. Depending on your budget, I'd get either an upgraded rear sway bar (Addco or something similar), or F&R swaybars. I run Bilstein HDs in my truck, and I've been pleased with them. Tires on my truck are 9 year old Firestone Destination ATs, which are positively awful (not to be confused with the new Destination ATs which get rave ratings on TireRack.com), so I'm sure I could gain a lot there by putting on some better rubber.
If you could only change one thing, an upgraded rear sway bar would be my recommendation.
