Learn Russian: Ты любишь этот старый фильм?

The dictionary form of this verb is любить (to love/like). It belongs to the second conjugation group because it ends in -ить. To make the ты (you, informal) form, we drop the -ить and add the ending -ишь, giving us любишь.

Doesn't любить mean "to love"? Why is it translated as "like" here?

Yes, любить is the direct translation for "to love". However, Russians also use it very often to mean they strongly "like" general things or activities, such as movies, music, or food. While you might already know the word нравиться for "to like", using любить for hobbies and general preferences is extremely common and natural.

Since "this old movie" is the object of the sentence, shouldn't этот старый фильм change its endings for the accusative case?

It actually is in the accusative case! Because фильм is a masculine, inanimate noun (it is an object, not a living person or animal), its accusative form looks exactly the same as its dictionary form. The words describing it, этот (this) and старый (old), follow the same rule and do not change their endings here either.

How do we know this is a question if the word order is exactly the same as a regular statement?