Thanks for all of those that have responded. A couple of clarifying points:
1. My international travel experience is pretty limites at the point -- Australia for 6 months and short trips to Ecuador, London, Paris, and neighboring NA countries. However, I am traveling to Singapore, Malaysia, and the Philippines in March as well as my week in UAE in May.
2. Having not experienced non-Western cultures (outside of Ecuador?), this is something that is really intriguing to me. Based on airmotive's comments, India would be the perfect place to push my limits. I guess here the question is, am I ready to experience this?
3. Looking to keep this trip fairly reasonable -- less than $1,000 for the week including positioning flight from AUH/DXB (maybe a little more if I go to Italy). I'm looking for middle market hotels/ground transportation (US standards), is this budget sufficient?
4. Thanks to deniah for bringing up the visa requirement for Vietnam. Something I had not considered.
5. For India, I discussed with a colleague and he suggested Delhi->Leh->Srinagar OR Delhi->Jaipur->Jaisalmer->Jodhpur (are one of these the triangle?) since it will be cooler this time of year compared to having my base in Mumbai.
6. For Vietnam, the only information I really have is what I've found online with suggested routes/schedules -- starting in Ho Chi Minh City and moving north.
7. For Italy, the big draw here is the history of the country and the ability to visit Vatican City. Not to mention it might be a little more comfortable surroundings if I'm traveling by myself.
Maybe I'll have a better feeling for what I'm looking for after I spend 2 weeks in Southeast Asia next month.... And I am a US citizen.