• On October 3, Paul VanKoughnett talked about p-divisible groups.
• On October 10, Johan Konter introduced abelian varieties and sketched the Honda-Tate theorem classifying abelian varieties over finite fields up to isogeny.
• On October 17, Johan Konter classified abelian varieties over the algebraic closure of a finite field.
• On October 24, Joel Specter gave some examples of abelian varieties over C.
• On October 25, Joel Specter talked about the relationship between abelian varieties in characteristic 0 and abelian varieties over finite fields, and gave an example involving counting points on a Fermat curve.
• On October 31, Philip Egger talked about level structures.
• On November 5, Dylan Wilson talked about Lurie's realization theorem and defined Deligne-Mumford stacks.
• On November 7, Dylan Wilson continued to discuss Lurie's theorem.
• On November 12, Rob Legg defined polarizations and complex multiplication.
• On November 14, Paul VanKoughnett constructed the Shimura varieties used by Behrens and Lawson.
• On November 19, Paul VanKoughnett constructed TAF.

References and other resources:

• General: Doug Ravenel's TAF seminar. Tyler Lawson's survey paper.
• Chromatic homotopy theory background: The Northwestern pre-Talbot seminar. The 2013 Talbot workshop. Jacob Lurie's course notes. Mike Hopkins' course notes.
• p-divisible groups: Demazure's monograph. (All the stuff on this website might be worth a look.) Messing's book (more general and more schemey than Demazure). Tate's paper. A course by Ren\'e Schoof on finite group schemes.
• Abelian varieties: Mumford's book Abelian Varieties is the canonical reference. Milne's course notes. The Honda-Tate classification; notes by Kirsten Eisenträger and Frans Oort.
• Level structures: The canonical reference (for elliptic curves, at least, though the general situation isn't really harder) is the first few chapters of Katz-Mazur, which doesn't seem to be available online.
• Lurie's theorem doesn't have a published proof, but statements and discussions of it can be found in chapter 7 of Behrens-Lawson, and in Goerss's TMF notes.
• Stacks: If you can read French, Laumon—Moret-Bailly might be worth looking at. If you can read encyclopedias, there's always the Stacks project. Anton Geraschenko has some notes for a course by Martin Olsson. My favorite reference, for the homotopically minded, is Goerss's notes on the moduli stack of formal groups, which teaches a lot about stacks through a single example that's central to chromatic homotopy theory.
• Shimura varieties: Behrens-Lawson's main references are The Geometry and Cohomology of Some Simple Shimura Varieties by Harris and Taylor; p-Adic Automorphic Forms on Shimura Varieties by Hida; and "Points on some Shimura Varieties over Finite Fields" by Kottwitz (the most accessible, which isn't saying much).
• The Serre-Tate theorem: This article by Katz is the best source. Akhil Mathew has a good discussion on his blog.