This Monk tune is typical of his genius/insanity. The melody couldn't be more simple.  There is a 4-bar melody that is played twice in C concert.  Then it is played up a 4th in F, and back down to C.  Then the whole 16 bar pattern is repeated -- except that this time, it is truncated to 14 bars followed by a simple 2-chord vamp.
That melody can be learned in a few minutes.  The hard part is the chord changes.  The melody points to simple, predictable chord changes, but not with Monk.  He aggressively re-harmonized these chords.  That's no problem when playing the head, but it creates a real challenge for improvising because you need to solo to the changes Monk wrote, not the changes anybody else would have written for this melody.  Take the time to really absorb these changes.

Here is a quintessential Coltrane tune.  The melody reminds one of "Giant Steps", but without all the giant steps.  That is to say, the melody is mostly based around the major 7ths and 9ths, which gives it that cool jazz sound, but the changes aren't as angular and unfamiliar as "Giant Steps".  This is a good tune as an entry point to the mind of Coltrane.

It is a simple 32-bar AABA song form, so that part goes very naturally.  The bridge has a very ordinary set of changes.  The A section is less ordinary, but not off-the-wall.  Spend a little time at the keyboard assimilating the chords in the A section.  That will be time well spent.

There is a long tradition of songs from the movies becoming jazz standards.  This is probably because once the songs become familiar to the public, jazz artists find the public more receptive to hearing those songs with a jazz treatment.
There is similar phenomenon where children's songs have become standards.  For example, "A Tisket, A Tasket", and "It's Not Easy Being Green".  In the present case, we have a song from a children's movie, "An American Tail".  "Somewhere Out There" has been performed by artists from Mickey Dolenz of "The Monkees" to Linda Ronstadt, Kenny Loggins, numerous jazz big bands and many small combos.
Here is an arrangement with a smooth jazz feel.  The song is AABA form with few complications.  Near the end of the B section, you find a single 2/4 measure.  Other than that, it is a rather normal flow with a small extension on the last A section.
In this arrangement, after the choruses, we have includes a little 4-bar vamp for some extra fun.