Pros
For graduate students and postdocs, who your advisor is will be the one factor that determines your happiness, so take time to choose wisely. (Of course, that's true for anyplace.) There are tons of smart people around, so you can always find collaborators or people to bounce ideas off of. There are many opportunities to check out seminars / events outside of your specialty. Yale has made some effort to increase professional development, at least for staff. For postdoctoral trainees, there is a postdoctoral association and other groups related to career networking (CNSPY or Biomed Career council), and outreach (Science Diplomats) you can join. There is an office of postdoctoral affairs which may be helpful depending on the situation. I have met some former postdocs who transitioned into careers in administration, and for those types of jobs Yale will often promote from within.
Cons
Some labs or research projects may be unstructured, or roles not clearly defined depending on the lab. Some labs are pressure cookers (shouldn't be a surprise at a place like Yale). There is a lot of bureaucracy dealing with grants, probably similar to other research universities I'm sure, but sometimes it's out of control. The office of career strategies (where you go for help with job searching) is currently undergoing restructuring and it seems postdocs and grad students are low on the totem pole. Of course, undergrads are the ones paying tuition in part for that but hopefully there will be staff hired who will be dedicated to these populations.