PATIENCE: I certainly think that this trait is overlooked quite frequently. Days can be stressful and clients can certainly be unpleasant, even when you are trying your absolute best. Having patience is extremely important in order to get through the day and to keep your job running smoothly.
ORGANIZATIONAL SKILLS: You might think of this as keeping all of your paperwork together, but I also mean making sure that you are organized in your thinking as well. Being physically organized is important but when it comes to mental organization, try to make sure you are also developing this skill. A lot can be going on at once, making sure that you write down your tasks and are keeping proper order of everything going on will certainly be beneficial in providing your clients and patients with the best treatments.
COMPASSIONATE: It is important to be compassionate not only for your patients but also for your clients. When a person's pet is sick is can certainly be stressful and scary. Making sure that you are compassionate and understanding of their situation is essential.
I could go on and on about other traits that are important in becoming a veterinarian but I think that the most helpful thing would be to gain hands-on experience with a veterinarian yourself. Seeing first-hand what veterinarians go through and how they handle different situations will be important. So while you pay attention to the medical side of working with a vet, also make sure that you are really hearing and analyzing how different veterinarians handle the situations that they have been put in. Each situation will require you to employ different sets of skills. Being put in these situations will be helpful in allowing you to develop these necessary traits.