Translation:I love my family.
The ommition of 的 is brought up at every available opportunity for some reason!?!? It's beneficial for a new language learner to learn the "most correct" use of the language before they learn what they can omit and why. Colloquialisms can slow down learning if they aren't in agreement with the standard grammar. (Especially important in studying Chinese because it's grammar rules are uniquely solid compared to other languages)
Yeah, you're right. Just "家" works. Although, it could also mean house/home.
To be honest, "家庭" is a better term to refer to "family" than "家人". It's more commonly used to refer to "family" in general while "家人" refers more to "family members/relatives" as it's literal translation is "family person".