It was undoubtedly the delightful songs for "Little Shop of Horrors" which made people sit up and take notice of Alan Menken. This had everything: giant alien weeds with deep voices, crazy sadistic motorbiking dentists (Menken's father was a dentist!), lots of 40s period Doo-Wop songs, and plenty of humour. Disney in particular saw Menken's songs having just the qualities they were looking for to support their plans for new musical animations. So they took him onboard together with his partner on "Little Shop of Horrors", lyricist Howard Ashman. Ashman and Menken had worked together before on the musical "God Bless You Mr. Rosewater", and Menken had been writing for the stage since the 1970s, but "Little Shop of Horrors" brought out his love of Rock and Roll and was an immediate hit first on stage and then at the cinema. The song "Mean Green Mother from Outer Space" also earned an academy award nomination.
The Disney partnership started with "The Little Mermaid" for which Menken also provided the underscore winning oscars this time for both the score and the song "Under the Sea", followed in quick succession by "Beauty and the Beast" and "Aladdin" (In addition to Howard Ashman, Tim Rice contributed some of the lyrics for Aladdin). It is the songs that mark the centre of Menken's contribution, but he has also been variously credited as music producer and/or arranger on some of these animations. When Ashman became seriously ill and later died, Menken continued at Disney working with other lyricists, treating us to "Pocahontas", "The Hunchback of Notre Dame" and "Hercules" not to mention several sequels (sometimes straight-to-video), spin-off TV series, stage shows, and various other adaptations (e.g. shows on ice).
Along the way Menken won yet more oscars for score and songs, and many of these songs are probably very familiar to the younger (and older) generations. Note that it is common practice for musical versions of films (and vice versa) to be adapted to fit the new medium with changes to the storyline and to the music. The Broadway musical version of "Beauty and the Beast" for example introduces a new song "A Change in Me" which wasn't in the original film. Here's an article in Variety Magazine, with Alan Menken interviewed by film music historian John Burlingame, and discussing the new-style Disney-Pixar animated musicals. " Tangled" is a modern retelling of the story of Rapunzel. The soundtrack includes some songs with this distinctly modern appeal (with lyrics by Glenn Slater), and some effective score composition with both well-represented on the soundtrack album: available at these links Amazon.com or Amazon.co.uk.
Of course the soundtracks of Menken's Disney musicals are widely available. Here are some links for the UK and US Amazon sites:
In addition you will find a large selection of Alan Menken's Sheet Music titles for various instruments including the following for piano and voice (and usually guitar chords):