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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.
TODO: What are the best archivers for each Amiga-like OS? Are they pre-installed? If not, where to get them.