From time to time I get inquiries about genealogical research in India. Yes, there are some records available online, but no they are not particularly easy to access and compared to the approximately 1.37 billion people who live in India, there are relatively few genealogical records that are easily accessible online. FamilySearch.org has some records but compared to the total population of the country, the coverage is not extensive. You can see a list of the available records from FamilySearch for the entire country on the FamilySearch.org India page.
Here is a screenshot of the Ancestry.com page for India:
If you look at the population of India and then look at the number of records in the collections, you can see that the coverage is very small. Probably, the best place to start is with the FamilySearch.org Research Wiki for India Genealogy.
One place to start looking for records from India is the National Archives of India.
The BBC ran part of a series on "Tracing your Asian roots on the Indian subcontinent." Quoting from that website:
Before The National Archives were set up in India, there was a system of keeping manuscript records which were created by Hindu rulers, sultans, Mughal and other Muslim rulers.
Records may be available in the form of palm leaf, bark, parchment, silk, leather and cloth manuscripts. Records created by the Indian local rulers, Hindu temples, Islamic shrines, gurdwaras, and waqf authorities (the Charitable Islamic Trust) may be kept at the local state libraries, museums and relevant State Archives.
These documents can give a certain degree of family history information on elite families and higher-ranking officials who served the local rulers.
In 1891 (during the British period), the National Archives of India was established as the Imperial Record Department in Calcutta. Since 1947, the National Archives of India has established four regional offices at Bhopal, Jaipur, Bhubaneswar and Pandicherry.The BBC article cited above is helpful in outlining where other records may be available.
There is an IndiaGenWeb page that can also link you to some resources.
There are additional possible resources on Wikipedia.org although the references will link you to other websites. There are quite a few records from British India, see "Online Databases & Records for Research in British India." Many of these British records are also available on FindMyPast.com.