I personally find Hartshorne's progression unsavory.
For chapter 2, you should really understand everything up to, and including, section II.7. This is all very necessary machinery to do anything rigorously with schemes.
Section II.8 is very nice, and is something that I would suggest learning before moving to chapter III, but perhaps not from Hartshorne. Up until that point (i.e. up until II.8) you have a wide set of tools to deal with a basic and important question: "when are two schemes isomorphic?" But, you lack one of the most intuitive, and useful of scheme theoretic properties--smoothness. Section II.8 lays the groundwork for being able to talk about when a variety (or more generally a map of schemes) is smooth, but kind of falls short (he doesn't really talk about, and not even very satisfyingly, smoothness until like III.10). It only discusses the case of when a variety over an algebraically closed field is smooth. This doesn't help you (even thought it will turn out to be a large part of!) determine when a map of schemes is smooth.
So, I would suggest reading II.8 from Hartshorne, because this gives you a fundamental geometric tool, which should come before the algebraic sledgehammer of cohomology. But, like I said, I would suggest learning about it somewhere else, for example reading chapters 21 of Vakil, and skimming chapter 25.
I would make a very similar statement for flatness, that I did for smoothness. It's not covered in II at all. It's give its treatment in II.9, again, somewhat unsatisfactorily. I would suggest reading chapter 24 of Vakil.
You can very safely skip II.9. While the idea of formal schemes, the idea of looking at 'formally close' neighborhoods of subschemes, is very, very important, it's not something you need to see on a first go through.