Assuming it's on career list, once you acquire and then train an advanced skill (2 advances), the next time you train (next rank), does it now only cost 1 advance each time you train it to increase 1 rank? (e.g., spend 2 to get 1 expertise die, 1 more experience to get a 2nd)?

I ask because when a non-career advance, it costs 4 and usually the non-career cost is x2 career cost.

It costs 1 advance to acquire an advanced skill (in your career skill list).

It costs 1 advance to train a skill (in your career skill list) whether it is an advanced skill or not.

Although it does cost 2 advances to acquire and train an advanced skill to rank one it is two separate 1 advance spends, not one 2 advance spend.

The cost to train a skill does not change no matter how many ranks you have and/or whether it is advanced or basic, so the answer to your question is 1 advance to purchase rank 2 (or indeed 3. or mastery).

Additionally acquiring a skill without training it is a valid spend, and usually what happens in our group; the ability for someone in the group to roll over no one with the ability to roll is very useful… especially when you need to know what kind of horrific monstrosity you are going to die to this session!