Even though I have been surrounded by Indian philosophy all my life, I am very new to the culture. Six years ago I started Bharatnatyam classes and found myself spending a lot of time around Indians and absolutely loving it.
At that time the only thing I knew about Bollywood was that people make fun of it. I had heard that the movies were long and overly dramatic and Americans I knew who had seen them rolled their eyes about them. But I was curious to see for myself, so I asked the girls in my dance class to recommend a first one to see.
They suggested Dil Se. I don't agree that it makes a good intro! It's very intense and very dark, and of course very long. I felt thrown in the deep end with that one. I would like to rewatch it now, since it's been several years and I've gotten a much better feel for Bollywood movies.
Now the term "Bollywood" refers to Hindi films being produced in Mumbai. There are also films in Marathi, in Tamil, and in many other Indian languages and those have their own following and are called other things. There are also movies known as "cross over." Those are movies that are either more Western in style or are filmed in the West. Mira Nair's movies are usually called cross-over (she did Monsoon Wedding and The Namesake).
I had seen Monsoon Wedding when it was in theaters, but that is far from a Bollywood style.
If people ask me now how they should get started with Bollywood movies, I recommend Swades.
I saw that one very early on too, long before I knew that the main actor in both Swades and Dil Se is one of the most popular heartthrobs, named Shahrukh Khan or SRK. I had no idea at all.
Of course, when I first saw Swades, I pronounced it "Sway-deez," which is pretty much the way that combination of letters would be pronounced in English. It wasn't until I watched it again recently and saw the title in devanagari letters in the opening that I realized it was "swah-desh" And from that I was able to understand what the title meant! This is a perfect example of why it is a mistake to write Indian languages in Roman script. :)
Swades is a good introduction because the story is easy to follow, it starts out in America so it's a good grounding for Americans trying to get into it, and the music is well integrated into the story.
After that, there are a lot of directions you could go. There are tragic epics like Dil Se or Raincoat (these aren't really my style, and I never ended up finishing watching Raincoat!), there are loads of romantic comedies like Kuch Naa Kaho, there are classics from the the 1950s like Pyaasa (which I have not seen) and Madhumati (apparently responsible for starting to spread the idea of reincarnation to the West), there are gangster movies and there are buddy comedies like 3 Idiots that came out earlier this year and I haven't been able to see yet.
The 1970s was, apparently, when the term "Bollywood" was coined, as India overtook America as the largest film producer in the world. There has been a lot of borrowing back and forth between Hindi cinema and American movies. Though probably more of a cross-over movie, Bride and Prejudice was a brilliant adaptation of Jane Austen's Pride and Prejudice to modern India. And going the other way, the director of Moulin Rouge said he was directly inspired by Bollywood.
Music has also had a great sharing between India and America. Both filmi music (the music in Hindi movies) and Bhangra have influenced American hip-hop and vice versa.
There have been some Indian style movies written and produced in other places with various degrees of success. Slumdog Millionaire was written, directed and produced in Britain, but was based on a book by an Indian author. Marigold was a 2007 attempt done in Hollywood. I watched it recently and found it fell flat, though I can't really put my finger on why. One fantastic attempt was a Direct TV miniseries called Bollywood Hero. It stars Chris Kattan playing himself. He's frustrated that no one takes him seriously in Hollywood and he stumbles onto an opportunity to be a leading man in India, so he takes off for adventures in creating a Bollywood movie. It's very funny and it's available from Netflix.
Netflix is a great source of Hindi movies. Much more so than Blockbuster and I switched from Blockbuster to Netflix for that reason. I found that once I got the feel for it, I loved the music and dance and the long stories.