How to normalize these tables to 3NF

I will not give you an exact answer for 2 reasons: 1) I'm too lazy to filter through all that text. There, I said it. 2) You wouldn't learn anything.

Third normal form is about having no transitive functional dependencies. In other words, if A determines B, where B does not determine A, and B determines C, you have a transitive dependency, so B and C could be put into their own table.

An example from your set could be the tables with city, state, and post code. In real world situations, zipcode can be used to determine city and state. Perhaps you could have a separate table with zipcode as the key, and city and state are the other two attributes. This can be a transitive dependency because address ID -> zip code and zip -> city, state as I said.

Another important thing to remember: if any fact appears twice, you can normalize even more. For example, [city A, State B, ZipCode C] likely appears multiple times, because I'm sure you have multiple people from the same area.

EDIT After looking this over and editing it I have found a lot more things to comment on, but as this is an assignment I will give you time to think about it and come back in a few days or longer to go over it again, if you're still curious.

EDIT 2

I will give you suggestions table by table, but will try to limit these to just pushes in the right direction.

staff - There are no transitive dependencies that jump out to me, everything in there is determined by the primary key, which is good.

staff_address - Why do you have a staff_id column? This is not a good idea. In addition - you have a transitive dependency with the address as I mentioned already.

vet and vet_nurse - These tables have the exact same columns, so why are there two tables? Certainly there's a way you can use one.

appointment tables - Initial appointment and follow up have the same columns. Again, there should be a way you can make them one. I will give you one direct suggestion for the appointment table: put date and time as one column, aptDate, and give it the DATETIME value type.

patient - The customer_ID value is in the patient table, so why should it be in the others? Also, previous appointments and previous treatments are going to be very difficult to track inside the database. You should see that as soon as you start inputting data.

product - As it is now, it doesn't seem too bad, but there are issues I will discuss later.

product_sold - Is sale_ID unique? If it is, how many products can be sold in one sale? This table is definitely not normalized well.

supplier - How many products can a supplier have? This is your hint at how to change the product table.

suppliers_address - Same issue with postalCode here. Also, why does suppliers_address point to supplier?

inventory - Aren't you already tracking all of these fields in the product table (except for price)?

These are potential problems I see, but I can't in good conscience give you solutions to your assignment. However, if this is one of your first attempts at a normalized database it's not bad at all.