I confess I have no advice to offer for Singapore, though it is a hop across from India, for I have'nt been there, so far!
However, India is a different matter altogether & would certainly need a bit of research, on your part, before you are able to cobble up a fine itinerary.
Apart from the normal, well known destinations, like Delhi, Agra, Jaipur, which constitute the Golden Triangle, you have some delightful rural properties far from the woefully crowded city centres. These destinations are the Fort-Palace Hotels - luxurious, superb locations, fine 'Shikar (Hunting) dish' menus & great ambience. Some of the finest that come to my mind immediately are:
www.shahpurabagh.com /
www.sardargarh.in /
www.rohetgarh.com etc. All these locate in Rajasthan. The best time for this destination is between October to early march.
The Foot Hills & Tea Estates is another area of great interest. The colonial bungalows with fireplaces, wooden floors & some of the most spectacular Himalayan sceneries, where ridge upon ridge of heavily wooded hills, culminate into eternal snows! The best time for such destinations would be 15 March onwards, till the advent of monsoons. Some of the most delightful properties include: The Cecils / Wildflower Hall (www.oberoihotels.com ) / Judges Court, Pragpur & the Glenburn Tea Estate (Darjeeling).
Tigers find their last refuge in India...still around with us, in despite of her hungry neighbours! The Corbett & Bandhavgarh offer great oppurtunity to see the feline race in all it's glory.
For the Slow Traveller these are havens, Far from the Madding Crowds!