Postdoc applications

Getting Started

First, see this timeline for applications by colleague Remy van Dobben de Bruyn.
Next, see this advice from Professor Lauren Williams.
Postdocs on MathJobs typically are listed as *** Assistant professor or *** Associate professor. Those without an adjective prefacing them are often tenure-track. If in doubt, check the job posting.
You can save job searches and receive email notifications by logging in to MathJobs and scrolling to the bottom of the page to enter your email address.
Not all institutions post job offerings on MathJobs.
Some people get job offers applying for a handful of postdocs, some apply for ~100. The difference is generally a matter of how many places you check-off on MathJobs.
Familiarize yourself with arXiv, LaTeX, and ways to include diagrams & figures in LaTeX (e.g. diagrams.sty, Tikz, TikzEdt, Adobe Illustrator and Bézier curves). I've found Detexify helpful too.
Interfolio offers a service called Dossier Deliver that will electronically deliver reference letters for you.

Attend conferences and workshops such as those listed in Academia Resources: For grad students to familiarize yourself with people in your field as well as allow them to put a face to your name.
Once you have some research progress, email contacts made at these events to see if you can visit them to discuss math of interest to you both. If they run a seminar, they may invite you to speak in it. You can often get travel funding or grants from your institution, their institution, or your advisor. More funding opportunities are listed under Career Advice here or here.
Send an email invite to faculty/potential letter writers to attend your research seminar talk.
Now you have built up a connection with someone whom you can ask to write you letters. They know you, and they know your math. Ask for recommendations well in advance (say by early August) for Fall postdoc applications.

In the research statement aim for 1/3 on your past thesis results + 2/3 on future directions, or 1/2 on each.
A quick way to get the BibTeX code for a reference is through your institution's access to MathSciNet by finding the article, then Select Alternative Format >> BibTeX. (A subset of the given bib info may show up when you cut & paste & typeset it into LaTeX.)
The teaching statement could include:
- anecdotes illustrating your teaching practices
- a "10-minute window" into your typical discussion section
- comments/ratings from student evaluations indicating your teaching goals were met

You could update each cover letter with the name of the university you are applying to.
Include names of people closest to your research at the institution. You may need to look up faculty at the university.
Especially include faculty names with similar interests on the AMS cover letter for positions through MathJobs; this makes your application more likely to be read in more detail.
If someone is very closely related to your research at the institution, they might appreciate a quick email from you introducing yourself and letting them know you are applying.