10 days is pretty short for India, by the time you get acclimated to the timezone your trip will have basically come to an end.
Off the top of my head:
-As said earlier drink only bottled water but also eat nothing raw (fruit or vegetables), chutney, ketchup or any other sauces. Eat shit loads of eggs, the eggs there are extremely fresh and taste way better than anything you can get here. I would also avoid street food vendors altogether.
-Go to the Taj Mahal (obvious) which is only a couple hours drive from New Delhi. When buying tickets for the Taj Mahal if you can get a local to buy your ticket it will save you a lot of money. They charge $20+ to foreigners and only a couple of dollars to locals.
-If you buy something MAKE SURE YOU BARTER. A lot of the time you can get them to drop the price by 50% or more.