The Broadway Veterinary Hospital on 12th Ave. in Capitol Hill is fantastic. All of the staff I've interacted with so far has been incredibly caring and informative. They give you as much information as you want and they make sure you understand your options and the care they are giving your pet.
As noted by another review, the parking can be a bit of a challenge unless you get there really early in the morning. But, there is additional parking in the back and so far I've had no trouble finding at least one spot. If your pet is on a special diet you will also love that they have a store downstairs from the vet's office.