To change the order of groups Go to wp-admin >> doctors >> doctor group and change the names of groups make the last first group on top which you want to display in first. So on once all groups names are changed as per the your required order edit groups or categories again and add 2 at end of each groups slug and update so duplicate slug do not make any problem. Again edit group and copy your group name put that in slug area WordPress adutomatically short the first letters and add - between two words.
Now go to wp-admin >> doctors >> all doctors and assign correct doctors or staff members to correct groups.
How to re order the posts or custom posts like veterinary doctors?
Stay in All doctors area or posts area edit the post which you want to show first by clicking quick edit now change its publish date to most recent date so it can display on top. Now quick edit the post which you want to show after first post make sure you change its publish date or time previous time or date to the first post or doctor.
That's how things will be sorted nicely. You can also try WordPress re order plugin but i felt that is not working properly.
You can watch this video which explains how you can re order the WordPress posts and WordPress custom posts.
https://youtu.be/PMnciSHcvVc