There is no right or wrong about how quickly to finish this game. Really, it never "ends" - you can just keep curing pets for infinity once you unlock everything. Here is how I went through the game, which only took me a few hours.

Day 1
You're just getting started here, learning how the commands work and how to earn money. You're dealing with thousands of guinea pigs named Ewok :)

Day 3
You learn that when people pick up their pets that have stayed overnight, they pick up, pay then, and leave immediately. It's nice to get an influx of cash quickly like this. I was ready for the cat test by this point.

Day 4
I started building the cat house here. You start out with plenty of food so you don't need to worry about that for a while. The actual 'work' is done instantly, overnight.

Day 5
I have a cat house now and can afford to build cat house level 2 - which puts an additional room onto the base house. You only get 40 "minute" between visitors, which is barely enough time to run to feed the budgies and return.

Day 10
By now only the stables and therapy are left to build.

Day 12
All but the therapy building and training are done.

Day 13
Start the therapy building.

Day 14
Start therapy building level 2

Pet Vet Walkthrough

Paws and Claws - Pet Vet Review