Borough Market is all about food! If you don't know what to do on Saturday, empty your stomach and go to Borough Market. I'm sure you can be full (I mean very full) when you leave the market.
For me, their seafood session is the best, it has oysters, shrimps, fish and chips or seafood fried rice. They are all so yummy. The vegetable and fruit there are very fresh too, so don't forget to grab a cup of juice and buy some strawberry home.
As it is quite crowded, be sure you will need to buy and stand there to eat. This is kind of fun to be. It reminds me eating in Taiwan night market.
A video about Borough Market has been upload, feel free to watch it before you go.