Michael Crichton kind of set the standard with his novel Sphere, in which those called in are: psychologist, mathematician, zoologist, astrophysicist and marine biologist. The marine biologist had to do with the plot, so we could look to The Arrival, where a linguist is a core part of the team. Jack McDevitt took a slightly different approach with The Hercules Text, which included a priest, a psychologist, a "scientist," a national security person, and a few bureaucrats. The religious angle I think is interesting, though I personally wouldn't choose a religious person for the contact.

For me, I think we need a behavioral psychologist (to study aliens without words), an astrophysicist (to learn about their system and how they got here), a math genius specializing in quantum mechanics (to share what we know about the fundamental parts of the universe), a linguist (however they communicate, we can try to decode it), and a biologist (to study their biological forms and explain our own).