The question can be reduced to to be into sth.
herein, hinein kommen would be a literall translation of to get into, i.e. in takt kommen (or den Takt or den Rhythmus, but less commonly just rhythmus) or in fahrt kommen (note the lack of capitalization as per Duden, but not die Fahrt) are common idioms (to get into rhythm, to take up pace respectively). However nicht in den Film kommen would be understood to be underage for a movie presentation, more so than in english. in das Lesen kommen works, sounds clunky already, whereas in das Buch kommen is not common at all. Since we are talking colloquialisms, commonality is important.
The reduced form would be drin sein, which is rather colloquial, perhaps informed by the English in the first place. It's used like in the flow, in extacy.
See also im Bilde (I am in the know, but not part of the picture).
A related idiom is hinter etwas sein, steigen, kommen (to get behind), i.e. to understand, to see the other side, but more commonly to uncover (hinter ein Geheimnis kommen). I general, etwas hinter sich bringen, überstehen (to finish, endure) is a common idiom that might work for the action movie as well as the book.
To me, to get into implies the activity: To get into the reading, rather than the book. You could read a whole book in and out, or watch a movie start to finish, without focusing your attention on it, without building a frame of reference to explore, etc., not expecting anything positive in it.
An older form is "in etwas machen". Er macht in Aktien (to be a merchant of, invested in, into). Also "nach München machen" means to go to munich.
I'm into stackexchange - ich steh auf SE, bin bei SE, bin in der SE community, etc.
I got into SE - ich steh auf SE.