I am a sitar player, so I’ll give a few hints.
First of all, there is Indian classical and other kinds of Indian music. Let’s start with the classical tradition. A really good page about this has been put together by my tabla teacher:
http://www.chandrakantha.com/
Generally, Indian classical tradition is divided into two: Hindustani music from the north and Karnatik music from the south. I know Hindustani music the best. Hindustani instrumental music is descended from the vocal style and is based around raags or ragas, or musical modes with certain rules attached to them, which often embody a time of day, a season, or a particular mood.
There is a usual order to a piece, with it starting out in a slow alap movement, then a jor and often a jhala movement, which are increasingly fast. After this (which can take up to an hour), one starts with gats, in which the voice or the instrument is accompanied by drums (usually the tabla hand drums). There is a slow and fast gat movement, each centered on a tune, usually in a 16 beat cycle. Then there is an extremely fast accompanied jhala movement to end the piece. Karnatik music is centered around ragas as well, and many are the same. The music movements are different, though. Hindustani music is performed by voice, the sitar, the sarod (a guitar-like instrument plucked with a piece of coconut shell), the flute, the sarangi (a fretted bowed instrument), the santoor (like a hammered dulcimer), the shenai (oboe), and a host of others. Karnatik music uses most commonly the voice, the violin or a veena, which is like a big sitar played with a knob slide, accompanied by other kinds of drums – often pakhawaj or mridingam.
Good Hindustani classical artists:
Ravi Shankar – sitar, played in a unique style descended from his training by Allaudin Khan, a sarod player.
Ali Akhbar Khan – sarod, son of Allaudin Khan
Vilayat Khan – considered master of the sitar, his style predominates among most young artists (as many of them are related to him – Rais Khan, Imrat Khan, Shahid Parvez, Shujaat Khan, Nishat Khan, etc.)
Hariprasad Chaurasia – flute, reinvigorated the flute as a classical instrument
Shivkumar Sharma – santoor
Nikhil Banerjee – sitar, of the “gayaki” or singing style
Zakir Hussain – predominant tabla player, son of Alla Rakha, who accompanied Ravi Shankar a lot, won a few Grammys if memory serves.
Imrat Khan – brother of Vilayat, sitar and surbahar (bass sitar)
Pandit Jasraj – voice
Karnatik violin – I really like Lalgudi Jayaraman.
There are a number of outgrowths from classical music. There is film music, often composed from a classical style (check out Ravi Shankar’s music for “Pater Panchali”), there are dhuns (music based on folk songs), there is light classical or thumri, and a bunch of other stuff. The Indian film industry is huge and it is a huge source of music. A.R. Rahman has been one of the most influential film composers. Vocal artists hired to sing over actresses’ dance numbers (“playback singers”) like Asha Bhosle and Lata Mangeshkar are also widely popular. There is of course also Hindu and Muslim devotional music. Perhaps the most familiar of these is Qawwali, which is Sufi devotional music made popular by Nusrat Fateh Ali Khan.
Overlapping the film music is Indian pop. Much of this is in a genre called bhangra, which to my understanding is mostly dance music, often from films. There are a bunch of really popular bhangra people out there, all of which are escaping me right now except Panjabi MC, who has that song with Jay-Z (based on the Knight Rider theme) and Daler Mehndi, who used to drive a cab in L.A. if memory serves me right.
There is also Indian fusion. You mentioned Sheila Chandra. You can also look for Shakti, which features some top-notch classical musicians (Zakir Hussain, L. Shankar, Mandolin Shankar, Hariprasad Chaurasia) with John MacLaughlin (the world’s fastest guitarist). There is also Ry Cooder and Vishwan Mohan Bhatt’s amazing album “Meeting by the River.”
From there is a pretty active field of Indian techno and house/lounge music. Try artists like Talvin Singh (his Anokha: Soundz of the Asian Underground is a good sampler disc of a bunch of artists), Thievery Corporation, and all the Nusrat Fateh Ali Khan remixes out there. Dan the Automator has a few discs of Indian film remixes (“Bombay the Hard Way: Guns, Cars and Sitars”, think brownsploitation music.) Also Karsh Kale, DJ Cheb I Sabbah, and Asian Dub Foundation.
I hope that is a pretty good start.