Foreign women will almost certainly experience India as sexist, but if you are confident, relaxed, and assertive, you are unlikely to experience any serious hassles. That said, traveling solo is only for the very brave and thick-skinned, unless of course you're traveling in comfort and have hired a car and driver for the duration (using public transport is when you are at your most vulnerable). At best, you will experience being stared at intensely for an unbearable length of time, at worst you may be groped -- some men are convinced that all Western women are loose and slutty. To a great extent, Western cinema and fashion trends have helped fuel the legend that women from abroad welcome these attentions, and you'd do well to take precautions, like wearing appropriate (modest) attire. On trains, on buses, and in other public places, you are best off ignoring advances or questions from suspicious-looking men. Another strategy that often helps single women travelers ward off unwanted male attention is to wear a ring and invent a husband; if you're approached, say that you are meeting your "husband" at the next station/destination. You should have little difficulty determining when a line of questioning is likely to lead to problems. In particular, steer clear of men who have been drinking alcohol. "Eve-teasing" (the word denoting unwanted attention and public harassment by men) is an offense in certain parts of India, and you are within your rights to report inappropriate advances or remarks to the police -- the easiest response, however, is to loudly tell the offender off, and even strike him -- you will almost certainly be supported by those around you. You may want to ask whether or not your hotel offers a special room for solo women travelers; these are now offered in a few upmarket hotels in the larger cities, and include special privacy/security features.
Note that women are excluded from entering certain religious sites and attractions (which we have pointed out wherever relevant), but this is unlikely to impact too strongly on your plans. Menstruating women are, technically, not entitled to enter Jain temples or mosques.
