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).
It has been nicely explained by sciencesolve. I have attached the diagram depicting the process.
Amoebas obtain food through the process of phagocytosis. There exist 5 steps in the process: ingestion, digestion, absorption, assimilation, egestion.
Since an amoeba has no mouth, it ingests its food with the aid of pseudopodia, small extensions used to ensnare food particles. These particles are then enveloped in vacuoles for transportation.
The digestion stage in amoeba occurs in the presence of several enzymes that break the large molecules of food into smaller and soluble molecules.
In the absorbtion stage, the small and soluble molecules of food are absorbed in cytoplasm, under the form of lipids and glycogen.
In the assimilation stage, the food deposited in cytoplasm is used to produce energy and repair cell function.
In the egstion stage, fecal pellets are formed around the digested matter and excreted from the cell in vacuoles.