My brief understanding was that we simply went there, stomped out their military, then left. That's like how the first few months of the 2000s war went, but it didn't work out the same way.

That's the issue, that's not how the 2003 war developed at all! In 1991 the Coalition did what it set out to do, ensured Kuwait's ongoing safety and inflicted a serious defeat upon the Iraqi forces. Then, they broke contact. The ceasefire eventually enforced a no-fly zone, ensuring Iraqi fixed wing aircraft could not effect a presence, but what was left of their military and their rotary-wing fleet were allowed to remain. There was no regime change.

The Bush Senior administration refused to get involved in an occupational role, and Saddam's government was able to crush Kurdish uprisings thereafter. An insurgency developed, but not against the victorious coalition, but rather the weakened Saddam regime. The drawback of course was that, due to the 'hands clean' coalition approach, the uprising was given support only morally and any material was only spoken of, and never provided.

This is in stark contrast to the Bush Junior administration, which after their thunder-run through Iraq in 2003, stayed to effect a regime change. The regime change necessitated an occupation, which in turn led to secretarian concerns being raised (e.g: Would the majority/minority be duly represented) and general anti-American sentiment manifesting in an insurgency. However I'll break off the answer here as this is in gross violation of our 20 year rule. I will not be taking follow-up questions on this either, for that reason.