James White’s Sector General books are full of first contact stories, and are set on a multispecies hospital and a rescue/ambulance ship. In fact, it becomes the Federation’s primary source of first contact, generally by rescuing and healing some unknown species.
Anne McCaffery’s Decision on Doona is a first contact story between a small colony of humans who discover a village of foxlike aliens; I can’t go into more details without spoiling, except to say that nothing is what it seems.
Both Nor Crystal Tears and Sentenced to Prism by Alan Dean Foster are first contact stories set in his Humanx universe. The first records the first meeting between humans and thranx, the second involves a stranded troubleshooter falling in with some silicon based aliens who stumble upon him.
Illegal Alien by Robert Sawyer involves a group of aliens called the Tosok who land on Earth and ask for help repairing their spacecraft, in return for some technology; everything goes smoothly, until a human is murdered, and the evidence points to one of the Tosok. He’s put on trial, and complications ensue.
The Gentle Giants of Ganymede by James P Hogan recounts the first encounter between Earth and the Giants, complicated by the fact that they have been travelling for 25 million years ( 20 years subjective ) thanks to a damaged drive, and their world is now the asteroid belt and Pluto. It’s the seocnd in a series, however.
Code of the Lifemaker, also by Hogan is the first encounter between humans and sentient robots derived from a defective alien probe/factory that has turned Titan into a robotic ecology of replicating, evolving machines.