If they start coming too fast you can click the podium and select disallow new customers to stop the flow and let you catch up. When you start out they will trickle in but this is ideal as it gives you chance to get used to the mechanics.
Once you have the clinic you simply need to open it up and patients will begin to appear. These menus work in the same way as restaurants but there are fewer things you need to adjust. Once you've purchased the clinic the default options for setup are absolutely fine but you can change your price markup, uniforms and treatment quality by clicking the edit icon (the wrench and store icon). Related: 20 Awesome Things You Had NO Idea You Could Do In The Sims 4: Cats And Dogs You could also buy a plot of land and build after purchase if you prefer.
Once you've built or downloaded a suitable clinic to purchase it simply click the cash register icon in the bottom right corner and select buy a clinic. There is a seperate waiting room, clinic and bathroom and some extra touches like chairs, a coffee machine and some food bowls and decor. The image above is my starter clinic which included the high quality version of the vet podium, exam table, medicine crafting table and medicine vending machine.