Python for economists:
Here are some very useful links that helped me migrate from Matlab, Mathematica and Stata to Python.
Getting started
Steve Byrnes has a very nice post on how to set up Python. He also describes how to speed up your code with Cython.
Learning Python from scratch (for economists only)
John Stachurski has some excellent lecture notes on computational economics that helped me learn Python. Once you go through all the notes, you'll be able to code in Python most of the things you can code in Matlab and Mathematica.
(update_1) The link above is now redirecting you to his new lecture notes (co-authored with Thomas Sargent).
These notes are simply superb! These are going to change quantitative economics in the same way the introduction of hindu-arabic numerals has revolutionized Western Mathematics. The new Liber Abaci!
(update_2) I've noticed that the notes don't mention anything about Spyder. It's probably the best IDE for people who are transitioning from Matlab. Winpython and Python(x,y) have it included in their installation pack, so there is no need to worry about how to install Spyder on your PC. If you have no prior exposure to Matlab, I urge you to follow John's instructions on programming using IPython and your favorite text editor.
Attend a PyCon event
(update_6) I had the chance to attend PyCon this year and learned a lot of new things about Python and its users. More on my experience can be found here (along with some links to the best tutorials and materials, all IMHO).
If you can't make it to PyCon next year, try PyData and/or join one of the Python groups close to you. The community is the best asset that a programming language can offer you. And, just to show you how awesome the Python community is, check out the latest videos at pyvideo (including most PyCon 2015 videos here).
From Stata to Python
Alex Bell wrote a great booklet on how to learn Python after being exposed to Stata.
Check out his sections on text processing and web scrapping. These are good building blocks for learning how to parse.
For NAG aficionados
The guys at NAG have a collection of bindings that call NAG routines in Python. Mike Croucher contributed to making this possible. You should check out his invaluable blog and sign up for his RSS feed!
(update_4) They started taking Python seriously (it was about time!). Check their blogposts on how to use their nag4py module. This looks promising!
For Dynare aficionados
Dolo is a nice substitute for Dynare. The goal of the people behind Dolo is to create a program that could easily separate between the symbolic and the numerical parts of the code. The user can then specify a numerical routine from a growing range of possible candidates.
Parallel/Cloud computing in Python (for advanced users)
For parallel computing in Python, check out the documentation on IPython. For running your codes on a cloud, check out pythonanywhere. However, before paying for such services talk to your colleagues at the Math department. You'll be surprised by how willing they are to share their computing capabilities with you (conditional on having a Python script).
(update_5) Google joined the cloud crowd. Prices are about to drop dramatically...
(update_7) Microsoft also started providing cloud computing services. It took them a while... (no surprise here)
Prices are about to drop dramatically
Visualization in Python
There are many nice ways to plot data in Python. The most commonly used package is matplotlib. While it's the best tool for fast work, I found that the seaborn is more visually appealing. Altair is another great addition to your toolkit (especially when it comes to your scatter plots).
Kudos to all the people above for making my life easier. Also, check the codes on my Research page for some unqualified attempts at coding in Python.
(update_3) For those of you on the Job Market this year who plan to send 200+ applications, I wrote down a little Python script that sends all your job applications to a list of prespecified emails. It also updates your cover letter with all the information you need for each employer (i.e. name of the institution and title of the job). You can find the codes below.