What is the requisite knowledge in logic required to study forcing?

Solution 1:

The fastest route to forcing is probably Nik Weaver's Forcing for Mathematicians. It gets you from the definition of an ordinal to the independence of $\sf CH$ in about 50 pages.

I would suggest using a different text than Jech for forcing. He uses Boolean algebras, which are sort of deprecated nowadays (Jech himself sort of abandons them rather quickly), and his treatment is very terse. Kunen's Set Theory would be a better introduction. I think the preface (of the 2013 edition) tells you exactly which chapters you need to read to understand the chapter on forcing.

There's a very new (2021) book by Mirna Džamonja called Fast Track to Forcing, whose title seems very relevant, but I haven't read it and thus can't comment on it.

To actually answer your question, you can happily skip chapters 7-11 in Jech (this refers to the 3rd edition). Chapter 12 is mandatory reading though, since that's where absoluteness is introduced.

For uses of forcing which require large cardinals or finer combinatorics, you might need to go back and read some of the stuff you skipped, but I think the above is sufficient to understand the basics. Still, I would once again encourage you to look at Kunen instead. If you must read Jech, having a copy of Bell's Set Theory: Boolean-Valued Models and Independence Proofs might be a good idea, as he gives a lot more details than Jech does.