What is good collection of these Indian myths? Is there one akin to One Thousand and One Arabian Nights; with more focus on India?
This one is "Weaver as Vishnu" on page 46.
Note: the HTML version is in Sanskrit, the PDF in English.
Edit: Of course, there are many other works of Indian mythology. The big two being the Ramayana and the Mahabharata, which if you consider it a single work, is the longest text ever written. The heart of the Mahabharata is the Bhagavad Gita.
To your question - the most popular ones are The Ramayana and The Mahabharata. Jataka Tales as someone has mentioned below. Also, Tales of Tenali Rama and Vikram & Betal are good reads too.
While this doesn't fall under the category of myths, perhaps one of the most comprehensive works on finance, war strategies, and international relations (yes, this from ancient India) is Chanakya's Arthashastra. Another book by Chanakya, Nitishastra is about the ideal way of life. Look up Chanakya on Wikipedia.
My suggestion would be to start with the Mahabharata, it is a whole literature in itself and is at par with some of the best in the world.
Start with a summarised version(The version by C Rajagopalachari is quite popular) and if you like it you can explore it in-depth with the version by van Buitenen or Kisari Mohan Ganguli.