(Εγώ) θέλω καφέ. = I want coffee.
In this sentence εγώ is the subject of the verb and for this reason it is in Nominative. Εγώ is usually omitted, because it is indicated by the verb ending (-ω). Καφέ is the object of the verb, that's why it is in Accusative Case.
The phrase Μου αρέσει ο καφές has a completely different structure in Greek. (This is not obvious at all from the English translation "I like coffee", and this is where the misunderstanding comes from.)
Μου αρέσει ο καφές. We translate it as "I like coffee", but the word to word translation is "The coffee is pleasing to me".
ο καφές is the subject of the verb and for this reason it is in Nominative Case
μου is the object of the verb (indirect object in Genitive case)
It is very important, when we study a language, to study the spesific structures of this language in detail and not just try to translate from one language to another. Syntactic rules are not obvious from the translation.