I didn't have too much trouble understanding the subject change with αρέσω, which is about the same structure as gefallen in German or gustar in Spanish.
But I didn't really pay attention which case is used. It seems to be genitive in the first couple of examples which use pronouns, and later there is σε + noun in accusative (basically the same structure as in δίνω το πουκάμισο στον άντρα, which I think of as "indirect object" though the term might not be used in Greek).
After reading a little about it, it seems that you have to use genitive with pronouns (i.e. στον αρέσει το πουκάμισο is wrong, it has to be του αρέσει το πουκάμισο) but with proper nouns both cases would be possibly, and σε + accusative may be more common, did I get this right?