Learn Russian: Анна должна чистить картофель и готовить суп. - Anna must peel the potatoes and cook the soup.

Должна means "must" or "obliged to." It is a short-form adjective, which means its ending has to match the gender and number of the subject. Since Анна is a feminine noun, we use the feminine form должна. If the subject were a man, it would be должен.

Why do чистить and готовить stay in their dictionary (infinitive) forms?

This is the main rule for expressing obligation: you use the correct form of должен followed directly by the infinitive of the action verb. It works a lot like English: "must peel" (должна чистить) and "must cook" (должна готовить).

What case are the words картофель and суп in? They look like they haven't changed from their dictionary forms.

They are in the accusative case because they are the direct objects of the actions (the things being peeled and cooked). Because they are inanimate (non-living) masculine nouns, their accusative form looks exactly the same as their nominative (dictionary) form.

I've learned the word картошка for "potato." What is the difference between that and картофель?

Картофель is the more formal, standard word for potatoes, often used in recipes, menus, or writing. Картошка is the informal, everyday word you will hear most native speakers use in conversation. Both are treated as singular words in Russian, even when we translate them as the plural "potatoes" in English.