Now, steering the subject back to the original question (and ignoring the Hummer vs Jeep for the purpose of this answer), there are 3 main things you want to worry about when offroading:
1. Ground Clearance/Center of Gravity
2. Traction
3. Recovery
A stock Jeep is very capable off the prodcution line, but that does not mean you cannot improve it. If I were to start again with a stock Jeep, the first thing I would ad would be recovery points front and back; small investment, but results in piece of mind that if you get yourself into a bind, you will be able to be taken out by another dirver. A winch is very nice to have, but a high lift can also be used to get yourself out of a bind with the same end result (albeit in a MUCH longer time!).
Ground clearance starts to get expensive the higher you want to get, and typically (there ARE exceptions, but not that many) higher ground clearance also results in a higher centerof gravity, which means it is easier to roll over (especially in off-camber situations). Ussually, you are OK up to 35" and dont have to worry too much about your CG being too high, and it is said that most trails can be run with 33" tires.
To clear 33" tires, typically you need a least 3" of lift (preferably 4"), but as you go to 4" lift, other things need to be modified, such as brake lines (need to be extended), Slip Yoke (you will most likely need a SYE kit and a new Drive Shaft as a result, but you can probably gt away with a TC drop - of course this will sacrifice ground clearance), and probably a modification of your rear axle (to correct the pinion angle).
As for traction, the first thing you will probably want is to look at new tires (I am talking about the thread pattern here, not the size); which tires will depend on what terrain you intend to navigate. Mud tires are ussually good also on the rocks, but tend to be noisier on the street, tend to dig in sand, and are not so good handling ice. AT tires are a good compromise between street ad mud/rock, and tend to do very well in sand and ice, but heavy mud will clog them up, and the carcas tends to be not as strong as the MT, which means they are more susceptible to tearing the sidewalls.
Then you go into the differentials; Jeeps come with open diffs, which means if you loose traction in one of the 2 tires, then the entire axle lost traction. Automatic lockers are a way to correct this, but they dont typically have good road manners (they can cause your Jeep even to change lanes in the highway if you are not careful, but you CAN get used to them and adjust your driving so that this is minimized to just a few clicks coming from the axle while you drive). Selectable lockers are (in my opinion) the best option, as they give you the option of Open Diffs for the road (which makes them invisible on the street) and a spool (which means the power is always sent to both tires in the same proportion, so offroad it will not matter if you have a tire up in the air, the other always has power as well applied to it). There are also Limited Slips out there, but these are more of a compromise (let's call it a lower cost, less efficient version of the automatic locker if you will). The problem when you ad a locker is that, especially if you have big tires, you will most likely need to upgrade your axles, as they are not designed to take the additional load (especially the rear if you have a D35; I am not sure which axle the 2009 JK Sahara comes with). I know the rubicon comes with electrical lokers, which are fine, but as far as I remember, the Sahara does not come with this option. Still, open diffs are not as bad as some make them sound, and if you have recovery points, even if you get stuck, you can be pulled out, so I would start with that, and see what else you need from there on...
And remember, how you build your Jeep will depend on what kind of terrain you plan to drive on. or example, rock crawlers have super low TC ratios to drive VERY slowly but with loads of torque; mud drivers need a little more wheel speed to clean the tires, so the super low TC are not as attractive as more HP. But the people running in sand will want even MORE HP and paddle like tires, while keeping the CG low; and the people running expeditions will ussually run smaller tires (around 31's typically) because it reduced the chances for breaking parts. So choose your type of terrain, and build for that type of terrain.
I hope this helps, and I hope you enjoy your Jeep!
Felipe