So, I was just looking and when you get a basic crafting book, it contains formulas for common items. My question is that some of the things you gain access to are formulas for alchemical items. How does this pair with the alchemical formulas from the class. i.e. If I learn lesser antidote from the basic crafting book, can I just add that into my formula book for use with the quick alchemy/advanced alchemy class features? Or is there something I need to do to copy the formula from one book to another so that I can take advantage of the class feature? Or is it completely separate?
The Alchemist gets a unique class feature, Formula Book, which specifies that they gain a formula book (item) and many of their class features reference that formula book, suggesting that you can only draw from that source.
The item in question also indicates
Formulas can also appear on parchment sheets, tablets, and almost any other medium; there's no need for you to copy them into a specific book as long as you can keep them on hand to reference them.
This suggests a rule that having a formula is the only requirement, not that it be in that particular book.
The bottom line
You're going to have to check with your GM(s) on this one. There are contradictory rules and it's not entirely clear if the Alchemist class features constitute "more specific rules" because they never use language like "only" or indicate that you have to copy them to the specific book (although the text does say "your formula book" in the singular multiple times). RAW it would seem that you have to copy them over because you cannot use your class features without that book, but RAI seems that you should be able to use any formula on hand for crafting including the Alchemists' expedited versions.
There may be a caveat that a GM could allow; the text in the Formula Book (class feature) is
You start with a standard formula book worth 10 sp or less for free. The formula book contains the formulas for two common 1st-level alchemical items of your choice, in addition to those you gained from Alchemical Crafting and your research field.
It does not specify that it needs to be a blank formula book. The Basic Crafter's Book is, in fact, a book of formulas and is worth 10s or less (1s, in fact). Oddly, blank formula books are (now? I couldn't find it in errata) valued at 1g and are technically not valid choices for Alchemists. This might suggest that the BCB is "full" and not a full sized formula book, but it's worth asking.
The rules on formulas have a bit to say about copying formulas from one formula book to another:
You can copy a formula into your formula book in 1 hour, either from a schematic or directly from someone else’s formula book.
But the more important question is whether you need to do so before crafting these items using your alchemist features. Many of the alchemist class abilities reference 'your formula book', and there's even a specific feature called Formula Book that grants you an initial book for free. Regardless, it seems reasonable that every formula book you own could be described as "your formula book", and there are some specific sections of associated rules that support this interpretation.
Quick Alchemy has a requirements section of "You have the formula for the alchemical item you're creating..." while also having an effect "You create a single alchemical item of your advanced alchemy level or lower that's in your formula book...". This seems to suggest that having a formula in a particular book and having a formula at all are considered equivalent for these features.
Finally there's the description of the blank formula book item that particularly supports this understanding:
A formula book holds the formulas necessary to make items other than the common equipment from this chapter; alchemists typically get one for free. Each formula book can hold the formulas for up to 100 different items. Formulas can also appear on parchment sheets, tablets, and almost any other medium; there's no need for you to copy them into a specific book as long as you can keep them on hand to reference them.