I'm starting to explore the possibility of investing in Indian stocks, but don't know quite where to start.
* Can you recommend any mutual funds that specialize in India?

Last time I looked, which was quite awhile ago, there were I believe only 2 open-end funds and 4 closed-end funds that specialized in India, see:
http://boards.fool.com/Message.asp?mid=12773574
I won't recommend any particular one as Indian funds tend to be rather volatile and you should look at the holdings currently of each to see whether they meet your needs and risk tolerance.

* If I wanted to invest in individual companies, can I do so with inexpensive online brokerages? Or must I go through a full-service broker? Any recommendations?

Again, last time I looked, Americans (with the exception of non-Resident Indians) could not invest directly in Indian stocks - only in ADRs/ADSs and perhaps GDRs/GDSs. ADRs/ADSs trade in the U.S. and you should be able to trade them with any broker (with some exceptions). GDRs/GDSs trade elsewhere (like London, Singapore, or...) and could only be traded with a broker who can handle foreign shares.
For more info on the ins and outs of ADRs, check out the sites of the main depositary banks at:
http://www.bankofny.com/adr/
http://www.citibank.com/corpbank/adr/
http://www.adr.com/

Best of luck,
David