After installing Anaconda, "~/anaconda/bin" was added to the $PATH variable. But now, I cannot run "brew":
-bash: brew: command not found
Only if I add "/usr/local/bin:/usr/local/sbin" to the beginning of $PATH can I get brew to work, but then I don't have the Anaconda python installation.
How can I have a default Anaconda python and and operational homebrew?
I think the problem is that your PATH is now mis-configured. What is your PATH and also the contents of .bash_profile? Ensure that you can access both via PATH with any mods in .bash_profile.
Also, I am wondering how many versions of Python you have installed? If the Anaconda installation came with a version of Python, then perhaps the version that homebrew installed should be uninstalled.
See Is path broken for anaconda ipython? and How to modify PATH for Homebrew?