The apostille process is both a state and a federal procedure, depending on the document. Apostiles are prepared by the SOS (Secretary of State).
Differnt EU countries dont have different requirements as far as the apostille, its a standardized procedure. Your state wants to know which country because many countries (non EU) countries get a differnt verification document. Some states put the name of the state on the actual Apostille but if not then the apostille for France is the same as the one for Italy. So you can DEFINATLEY start in advance.
Typically the most common documents you will need apostilled are your degree, teaching certificate and possibly your criminal background check. Most of these are at the state level, but for instance my DODEA teaching license goes to the US SOS, as well as my FBI criminal background check.
Many people are confused what an apostille is and think that it is authenticating a document and its not. All an apostille is, is verifiation that the signature and seal of a government officer (typically a notary) is valid (meaning the signature matches, and the commission or office of the official is correct and current).
The most common mistake when requesting an apostille is that the register of your school thinks that means you want a notarized copy of your degree. So the register takes a copy (usually one you provide for them) and then notarizes it, and thats incorrect.
To get an apostileld copy of your degree you provide the register with a copy of your degree with the statement on the back or in an attached dcoument (and this is important) attesting under oath that the document is a true and exact record. The actual language of the statement varies from state to state, and you may need to look this up on the internet, if your register doesnt know.
The register then signs this statement (which is actually an oath) and hs the statement notarized. Thats where the apostille differs from a certified copy. The register is swearing an oath that the degree is true and accurate and the notary is administering and witnessing that oath.
You then send this to the SOS and the SOS verifies the signature of the notary and that their notary commission hasnt expired. They then print out a very European looking document and attach it to the degree document from your register.
The process is nearly the same for your teaching certificate depending on your state. The difference is that the official who handless teacher certification in your state may not need a separate notary. They may have their own seal, and signature on record with the secretary of state, so you just need a "sealed and sworn" copy for an apostille and you send that to the SOS.
When it comes time to the US SOS, you typically make the request for whatever document it is from the managing agency and they forward it themselves to the US SOS for attach the apostille. For instance my FBI criminal background check goes directly from the FBI to the US SOS. My DODEA license goes from DODEA HQ to the US SOS.
Lastly, some countries and school want an apostille even though they arent . to the Hague Convention. In these cases your state SOS may print up something different then an apostille, and the school will insist on an apostille. In these cases you just need to put an EU country on the apostille application and give that to your school.