I am aware that Linux distributions usually have 'package managers' to allow me to download and install software from the command line, and that I should also be able to download and compile source code from the Web. But I am not sure about how best to do either of these things on the Raspberry Pi.
You need to be connected to the Internet.
If you haven't done so recently it is worth running,
sudo apt-get update
This updates the list of packages that are available to apt-get.
Then use apt-cache to search for the package that you want to install, in this case,
apt-cache search curl
Every package name or description with the word 'curl' in it will be displayed in a list. This is useful in cases where you are not sure as to the exact name of the package that you want to install.
Once you have the name of the package that you want, install it as follows,
sudo apt-get install curl
If you're not sure what you're looking for or just wish to browse through what's available, you can alternatively use the menu-driven tool Aptitude:
As MINImalistic raspBIAN is, at its heart, still a Debian derivative, the apt tools are the tools of choice for this really cool OS. Thus, almost anything said about Debian also applies to MINIBIAN, except, of course, since it is not only designed to be "mini", but in fact is minimalistic, there will be a lot of "Ign"(ore) messages while running the "apt-get update" command.