What food did people eat in Shakespeare's times?

About the same diet that people eat today: they ate meat and vegetables and bread.

Some things were different. They did not have new world foods like potatoes, tomatoes, corn, chocolate, and vanilla. Coffee and tea were not yet popular. Sugar was ridiculously expensive and not used; honey would be easier to get. There was no refrigeration so milk was difficult to keep. Recipes of the time called for almond milk (crushed almonds soaked in water) instead. Milk that was turned into butter and cheese kept a bit better.

On the other hand, the Elizabethans in some ways ate a more varied diet than we do. They ate a broader spectrum of meat, including the meat of swans, peacocks, and deer. They also had different vegetables and fruits. Fruits like quinces and medlars are unheard of nowadays. Salads (which they called sallets) included a number of greens and herbs we do not use as well as edible flowers.