I spent a few months backpacking around India in 2001. Given that the OP has only 10 days and is starting in Delhi, I'd suggest sticking to Rajasthan. My favourite was Jodhpur but Udaipur, Jaisalmer, Jaipur etc were all excellent too. Other options would be short hop trains to, say Varanisi and flying back - lots to see along the way. Or flying to Kerala for a week on the backwaters.
Getting around: I was on a budget and had oddles of time, so a driver/car was not something I considered. I had no issues with 2AC train travel, other than getting tickets, but the more memorable moments were spent in the cheaper carriages. I'm guessing that 1st class would be full of tourists - the natural enemy of the discerning traveller - or OS-based Indians. I found that being set upon by the touts for accommodation yielded excellent results.
Eating: I travelled for a short time with a UK couple who did the vegetarian thing and they claimed to be 'sick' more often than not but I wouldn't call the occasional squirts being sick. I ate meat every single day, often breakfast, lunch and dinner. To forgo the grilled kebabs, fried things, goat curries etc would be a shame IMO. The only time I got sick was in Varanasi where I ate tomato soup at a guesthouse. Sooo sick. Couldn't move for three days. I also drank a lot of the squeezed sugar cane, chai and fizzy drinks made on little carts (the carbonation process and little black marble 'pop' is fun to watch). I reckon a bout of illness is more likely to be brought on by iffy water, eating pakoras etc cooked in bad oil, or not washing your hands properly rather than meat itself (but by all means use some discretion when choosing a vendor). It'll be the first thing you chat about with other travellers.
