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The best sources for applications are:
Most Amiga applications are distributed as archives, typically lha, or alternatively tar.gz, zip or rar. To install the applications you need to unpack the archive (see list of archivers below). You will usually find a text file describing how to install the software or an “installer” script. If there's an installer script, just double click it and follow the instructions. You can choose wherever you want to install it, but it's usually best to have a separate volume (see filesystem) for your applications and data, and leave the system volume alone.
Command line tools
Shell (command line) archivers/unarchivers can be configured via filetypes, so actions (like moving or doble-clicking) with an archive file can have predetermined behaviours (like listing, unarchiving, adding, etc) under Workbench and most filemanagers. That way, users don't have to use the shell if they don't like to.
TODO: What are the best archivers for each Amiga-like OS? Are they pre-installed? If not, where to get them.