Is this imperfect or preterite?
Should this sentence be imperfect or preterite and how should I word it in Spanish?
"My family moved across the country many times."
5 Answers
It will be the imperfect, the 'many times' is the key as it make the time frame repetitive and vague rather than specific.
I find it helps if you think of the past in three ways:
1) PRETERITE - a dot in the past - specific
I ate breakfast. This happened today, Monday, at 8am)
2) IMPERFECT dashes in the past - repetitive, vague or non-specific
My family moved across the country many times.
Remember the 'many times' is the key. If you said '5' times it would be the preterite as you have suddenly taken it out of the realms of 'repetitive, vague or non-specific' and made it very specific with a number.
3) IMPERFECT a line in the past - ongoing past action
We were moving across (imp) the country when we heard the news (pret).
Perhaps this lesson might help. And you'll find a free podcast version at the same link if you prefer to listen when learning.

The wording can be many different ways.
Context determines imperfect or preterite, ie: if your family is deceased, then preterite (se mudó) would be used. If they are still alive, imperfect (se mudaba) would be used because there is always the possibility that they will move again.
Just an additional point - many times the imperfect will be expressed as "was (verb) ing" or "were (verb) ing". Another construction is "used to".
In your example that would be "my family was moving ..." Or "my family used to move..."
As Kiwi noted, the tip off here is the "many times" phrase, making this action a "broken line" in the past.
I also think Julian's comment is also true. Context is everything, and it can easily change the selection of tense.
Awesome answer by Kiwi. Yes, definitely imperfect, for all the reasons she listed. Fully in the past doesn´t alone call for the imperfect. These are tough for we new people.
Preterite, because action appears to have started and finished in the past.
Mi familia se mudó a través del país muchas veces.