Learn Russian: Мы любим планировать будущее вместе.

After verbs expressing likes or dislikes, such as любить (to love) or нравиться (to like), you almost always use the imperfective infinitive. This is because you are talking about enjoying the general process or habit of planning, rather than focusing on completing one specific plan.

Is the word будущее (the future) related to будет (will be), which we use to talk about future age in this lesson?

Yes, they share the same root! Будущее was originally an adjective meaning "that which will be," but it is now used as a neuter noun meaning "the future." So the future tense verb будет and the noun будущее are closely connected.

What case is будущее in this sentence?

It is in the accusative case because it is the direct object (the thing being planned). Since будущее is a neuter word and represents an inanimate concept, its accusative form is identical to its nominative dictionary form.

Why does будущее end with a double vowel -ее?

Because it grammatically behaves like an adjective. It uses standard neuter adjective endings, just like хорошее (good). Even though it means "the future" as a noun here, it keeps its original adjective spelling.