If you have not already done so, please invest in a book on FileMaker and/or take in some training tutorials. These will cover your questions in far greater depth than can be done in an online forum. Please feel welcome to keep posting questions, but if you avail yourself of additional training, it'll provide you with a better context from which to ask questions and understand their answers.
What tables do I need?
There's no simple answer to that question as it has several interlocking questions only you can answer. How do you run your vet practice? What are your current record keeping/billing methods? Do you want to design a system just like you have now, but on the computer or are there changes you want to make?
Be prepared to read up on database design, then sit down and map out what you think you will need on paper. Typically, each table can be seen as a list of very similar types of information. If you picture a bookkeeping ledger or computer spreadsheet, each "row" is a record and each "column" of information is a field. Some tables will be obvious, you'll need a table of customers, a table of pets. Others may not be so obvious such as a table that's a master list of all possible diagnoses with categories that make sense to you the vetrinarian. (What animal types does it apply to?, symptoms are typical?--just to name two possible ways to categorize them that I can imagine...)
Even more critical is mapping out how to "connect" your tables so that you can, for example, pull up a list of all the pets owned for a given client.
What if I wanted to choose from a list of diagnosis, for example?
That's called a value list and many different types are possible. You can even define conditional value lists where the values listed change depending on what you select in another field (Select "dog" in an animal type field and your diagnosis list can filter to just canine specific options.)
The options and implementation (custom list, table of values, conditional value list, pop up menu vs drop down list) are too varied to answer for you. I suggest you try different options and learn how they work so that you can select what works for you.