Which is the strongest revision of the following sentence? for her birthday, sharon wanted to do the following: eating at her favorite restaurant, go to the movies, and indulging in some fine frozen yogurt to finish off the evening.

What this sentence lacks is parallel construction. In order to achieve that, it needs to have the same form of words throughout (especially in this list of things Sharon did).
So the correct answer would be:
For her birthday, Sharon wanted to eat at her favorite restaurant, to go see a good movie, and to indulge in some fine frozen yogurt to finish off the evening. 
As you can see here, infinitives are used throughout: to eat, to go see, to indulge, and to finish off.

Go on google and it will show it