Answer:You could defend either choice in a sentence like that. Using gustan would certainly seem logical, and it is indeed said that way sometimes. But it is far more common to use the singular, gusta. It's kind of like shortening "me gusta la hamburguesa y me gusta el queso" by leaving out the second "me gusta," just as in English we might shorten "the happy children and happy adults" to "the happy children and adults." Why say "me gusta" twice if once gets the message across?
According to the Royal Spanish Academy, the singular verb should be used in a sentence like this when the two things you're talking about are uncountable or abstract and they follow the verb (as is usually the case with gustar). Here's an example the Academy gives: Me gusta el mambo y el merengue. Note how the two subjects are uncountable (they are both types of music). Here are some other sentences that follow this pattern:
- Es una red social de gente que le gusta el deporte y el ejercicio. It's a social network of people who like sports and exercise.
- Me encanta el manga y el anime. I love manga and anime.
- Me gusta la música y bailar. I like music and dancing.
- Al presidente le falta el coraje y la voluntad política para resolver los problemas de nuestro país. The president lacks the courage and political will to resolve the problems of our country.
- Si te gusta el cine y la tele, querrás pasar tiempo en California. If you like movies and TV, you will want to spend time in California.
- Me duele la cabeza y el estómago. Me duelen la cabeza y el estómago. I have a headache and stomachache.
- Me gusta mi cama y mi almohada. Me gustan mi cama y mi almohada. I like my bed and my pillow.
- A Raúl le gustaba el taco y el helado. A Raúl le gustaban el taco y el helado. Raúl liked the taco and the ice cream.

