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,
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,
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,
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:
Pacman is the package manager used in Arch Linux.
Synchronize Repositories and update System
Search for a Package
Install a Package
Uninstall a Package
Yum is the default package manager used in Fedora.
Update Installed Packages
Get Package Information
Install New Package
Remove Installed Package