none of your formulations are used. Usual ones would be:

“Nous sommes allés au cinéma, puis nous sommes allés manger.”

“Nous sommes allés au cinéma, puis au restaurant.”

“Nous sommes allés au cinéma, puis nous sommes allés dîner.” (plus rare)

“Nous sommes allés au cinéma, puis nous avons dîné [en ville, à la maison].”

In fact, this is the question you must ask yourself :
Do the 2 expressions share the same verb or not ?

For example,

We went to the cinema, then (we went) to the restaurant.

Would be translated as :

Nous sommes allés au cinéma, puis (nous sommes allés) au restaurant

Because they share the same entire verbal form. The ‘we went’ part is therefore facultative.

However, one should translate

We went to the cinema and had dinner.

into

Nous sommes allés au cinéma et avons dinné.

Because they do not share the same verb.
(you could find a better translation for the last one, but I took a litteral one)

So, in your case, the second translation should be used, although the same idea is more elegantly shared by the first translation.

The long (and correct) version is:

Nous sommes allés au cinéma et nous avons pris le diner.

A shorter (but still correct and more smooth) version is obtained by not repeating "nous":

Nous sommes allés au cinéma et avons pris le diner.

You can’t shorten the sentence anymore whithout changing the structure. "sommes allés" and "avons pris" are here verbal forms and can’t be cut.

Even if the two verbs use the same auxiliary, you can’t always simplify, for example :

Nous sommes allés au cinéma et nous sommes tombés sur ta soeur.

Cannot be shortened into:

Nous sommes allés au cinéma et tombés sur ta soeur

However it seems that there are a few exception, as sentences like:

Tu es parti et revenu.

Elle a triché et gagné la partie.

Sounds perfectly correct.