Nutrition

most tortoises are strictly herbivores, meaning they solely eat plant matter.

(example: Gopher tortoise's eat various types of grasses, wild grapes, blackberries & blueberries, etc)

many turtles eat an omnivorous diet, meaning they eat both plant and animal matter. 

(examples: painted turtle eats worms, algae, and plants, snapping turtles eat insects, worms, plants, small mammals, etc)

many sea turtles are gelatinivores, meaning they only consume invertebrates (like sea squirts or jellyfish)

example: leatherback 

many freshwater turtles are scavengers, meaning they eat carrion and refuse

 (example; eating dead fish) 

Digestion

food moves from the pharynx (back of the throat) down the gullet into the esophagus to the stomach. 

Food then gets passed to the small intestine by the pyloric spincter muscle,

 After the small intestine receives bile from the liver, the waste leads into the large intestine( also known as the colon) which then passes to the cloaca (a multipurpose cavity that secrets waste, eggs, and sperm).