The best place to start is his Sonata for Cello and Piano, the Double Concerto for Harpischord and Piano, and the Sonata for Flute, Oboe, Cello, and Harpischord, all collected here. The string quartets and the sonata for violin and piano are also important, plus the short late works for solo instruments or small ensembles or voice; choose by which instruments you like best.
Here is Q&A with Elliott Carter, interesting throughout. It ends with this:
What are the things that people say to you that are meaningful?
People say they like my music very much, but I’m 100 years old.