What is the best way of breaking a mathematical development into a sequence of articles? I am currently in the process of turning my math doctoral thesis into articles to submit to journals. This is my first time writing anything mathematical for publication.

My thesis was on the long-side, and referenced a lot of background material from areas of math that do not often communicate. (The principle field is topology, but it draws heavily on some specific corners of algebra, number theory and relativity theory that most topologists do not study.) It also has a historical component and the first result is a completion of a somewhat archaic classical idea. I then go on to develop that idea to obtain results pertinent to current active research.

Due to this, as well as the common advice of not making an article too long (especially one's first article), I am breaking my results into two articles. The first one will say something about what I intend to do in the second for motivational purposes, and lay all the groundwork, but is focused on generalizing a classical theorem. Then the second will focus on results more relevant to current topology, referencing results form the first but not weighed down by them.

My question is about what I should include in my setup at the beginning of the second article. I summarize a bunch of standard ideas at the beginning of my first article, which really does need to be there because it is not typical for someone working in any one of the fields to be familiar with the other concepts. I also introduce notation. Then in my second article I use all the same stuff, plus some more.

Is it okay to leave out a repetition of that background info, and refer the reader to my first article? I figure I should re-define any original or unusual notation, but I don't want to have all that repetition in the first section of both articles. On the other hand, I want people to read the second one, and not be off-put by being referred to the first one, or feel the background is not well established.

Is there a standard protocol for writing a series of two or more articles that develop a concept continuously, which draw upon the same set of background definitions and theorems? Would it be better to just write one very long article after all?
Each journal article should "tell a story". Break the thesis up into coherent stories. 

It looks like you're already doing that. The first article concentrates on generalizing the classical theorem, and the second gives the results that are relevant to current research.

I would suggest just putting the groundwork you need for the generalization of the classical theorem into the first article (unless you can save a lot of space by developing some of the groundwork for the second article at the same time as the groundwork for the first). And you should definitely mention the second article to provide more motivation for the groundwork. But putting groundwork that's not needed in this first article will detract from the "story". 

In the second article, you should include a quick review of the groundwork developed in the first article, and refer the reader to the first article if they want more details. Then add the extra groundwork needed for the second article, and  proceed to the results of the second article. 

One comment from experience: if you put a result that does not fit the "story" into a paper, when people read your paper, they are very likely not to even notice this result. So it's much better to put it in a separate paper. 

On the other hand, as you can tell from the comments here, many people consider it bad form to spread one story over two papers. So you should split a long result into several papers only if you can formulate it as several related "stories".