As you've correctly identified, there's several aspects to this - the vet but also the practice, the nurses and how it is run. A good practice will be well run and efficient, with welcoming reception staff. Good parking is helpful and a spacious waiting area, where ideally cats and dogs can be separated. Treatment rooms should be spotlessly clean, well lit, with accessible IT, and good temperature control (not sweltering in summer and warm in winter).
I'd say good vet nurses are as important as the vet - they carry out a great deal of the care, many procedures and have a lot of interactions with clients.
Admin/customer friendliness - accurate, clearly laid out bills. Giving you the right meds at the right price. Sensible opening hours that are clearly advertised. Available for contact via phone or email, answers phone promptly, emails also dealt with efficiently. Holiday opening times widely advertised. Appointments generally run to time (obviously subject to emergencies).
The vet - you should feel safe in their hands and be able to trust their professional judgement and practical skills. You should be comfortable in asking questions or asking for things to be explained. I can honestly say I trust my vet completely, and regard the relationship very much as a partnership with both of us having the goal of our animal's health and wellbeing. I want to know what I can do to support what the vet has done, in terms of home care and what symptoms/issues should be a concern, and what recovery should look like.
Our vet has seen us in the deepest emotional distress (having beloved animals PTS and dealing with serious emergency treatment) - both she and her team of nurses are empathetic, supportive and understanding. They helped us make choices about treatment options, and explained clearly about costs, benefits and risks.
As a client, I see my job as :- be on time, always pay promptly, be clear about what's wrong, when in doubt, ask. Finish the course of medication and obey instructions about aftercare. Treat vet staff with courtesy and respect. Give at Christmas and birthdays - it's not expected but it's the least we can do to say thank you to the people who care for very precious members of the family.