Amoebas take in nutrients via phagocytosis and pinocytosis. Phagocytosis is what occurs when an amoeba surrounds a solid food particle with its pseudopods to form a vacuole (known as a phagosome) in which digestion of the food particle then occurs. Pinocytosis is a similar process, but it has to do with the ingestion of a liquid via the budding of the cell membrane of the amoeba to make a vesicle similar to that formed during phagocytosis.
Once the nutrients enter an amoeba via the vesicles formed during phagocytosis or pinocytosis, then the nutrients within the vesicle must undergo digestion. This occurs when the vacuole fuses with a lysosome that contains digestive enzymes (for a more detailed explanation of such, please see the link provided below).