Translation:The whole family likes this restaurant.
It depends on whether you think the family in this case should be treated as a few people or a single unit. As a group of people you should use like and as a unit likes. To me it seems you are talking about a few people so it should be like (ie plural) if you said this family (ie one unit) I think it would be likes. See the following https://www.englishclub.com/grammar/subject-verb-agreement-collective-nouns.htm
The verb piacere works with the indirect object and third person verb so I like is written to me like - mi piace or you can say a me piace. If the thing that is liked is plural eg restaurants it would be mi piacciono or a me piacciono. In this sentence the whole family .. you say to all the family so you need the a - a tutta - restaurant is single so you use piace. You don't need the loro. If you said they like the restaurant you can say a loro piace....