If you're looking for an authentic shopping experience, you can do worse than the Night Market at the Botataung Paya. The Paya, located near the river in Southern Rangoon, is one of the city's grandest sights lit up at night. On the Paya grounds is a small Buddhist monastery, making this an excellent place to meet and great novice monks (many of them study English and are happy to talk to foreign visitors).
Available for purchase at the market are all manner of treats and sweets, including fresh sugar cane iced drinks, gold-leaf bananas, old Burmese coins and bank notes (it's impossible to understand how interesting a 3-dollar note is until you visit this place) and a bunch of other surprises I won't spoil by writing about them.
Few foreigners visit the Paya at night, making it an exciting place to interact with local folks. The excellent location south of Strand Road makes it an easy place to get to from anywhere in the city.