What do you mean by glue together points? Will this still be separated and irreducible?

I have edited my question to ask why any two points can be joined by a chain of rational curves. I agree if we're using your defition then it's obvious.

@Jack Why is it completely obvious as you say? For example if my points are outside of the image of rational map from projective space, I don't see why it should be clear I can get a rational curve connecting these points, or even a chain of curves.

@jlk Also in order for your assertion that the general section meets the fibers $f^{-1}(x)$ and $f^{-1}(y)$ from a dimension count, I believe you also need that dim $X\geq 2$. You don't just need it to define the blow-up, unless you had some other dimensional reason in mind, which I'd like to here. Thanks for the nice answer.

I'm familiar with Edidin's notes which are good, but is there anything more explicit than that?

@ulrich: I have Mumford's GIT, but does he actually include the details there? In the famous paper of DM, already on the first page they start quoting details from Hartshorne's Duality and Residues, which I haven't studied. I was hoping for something more beginner friendly, which included the details of why the relative dualizing sheaf is ample and in fact why is exists/what it is. I know now how to describe it explicitly but only after piecing together many different sources. Is there anything which actually explains the machinery necessary?

Sorry, now I understand, this is a linear subspace, and thus isomorphic to some affine space. Since affine spaces are irreducible any non-empty open set in them is irreducible (in particular principle open affines) and thus connected. Thanks for the suggestion.