"The restaurants like families."
Translation:Die Restaurants mögen Familien.
mögen is a verb, gern is an adverb. All "mag" verbs are used to say that you like SOMETHING; gern(e), being an adverb, is used to say that you like DOING something. For example: Wir mögen Pizza-- We like pizza. Wir essen Pizza gern-- We like eating pizza. Gern pretty much just means "with pleasure". (And there is no difference between gern and gerne. They're perfectly interchangeable) :)
"Mochte" is past tense ("liked") and used with the 1st person singular (="I") or the 3rd person singular (="he/she/it"). E.g. Der Mann mochte das Restaurant. = The man liked the restaurant.
(In case you meant "möchte" -- that means "would like" for the 1st person singular and the 3rd person singular, i.e. it's not the right word here either.)
In the sentence in the exercise, you need the present tense of "like" and the form for the 3rd person plural ("they") = "mögen".