The Amiga 500 has a very console-like experience for the most part when it comes to games. Treat DF0:, the internal drive, like you would the cartridge slot of a Super NES or a Sega Genesis. Just put your game into the internal drive and let the Amiga boot from it. :)
You can either leave the game in the drive and power the machine on, or alternatively you can power the machine on first, wait, and insert your game when the 'Workbench hand' appears. This screen doesn't literally mean you have to insert Workbench, by the way, it means you can insert any bootable disk. Almost all games and software will boot from this screen, when inserted into the internal drive.
For games that support multiple disk drives, such as The Secret of Monkey Island or Superfrog, first power off your computer, attach any drives you have available, then power on the machine. Insert Disk 1 into the internal drive and the game will start. Since Monkey Island comes on multiple disks, it will occasionally ask you to insert a specific numbered disk into any drive. You can eject and swap disks during gameplay only when the game asks you to. Many games won't recognise a second drive. It's not a hardware fault: they're simply not programmed to use it. If nothing happens when you put Disk 2 into the external drive when prompted in a game (wait several seconds first for the disk change to be acknowledged, or try pressing Fire), try the internal drive instead.
There are relatively few games that run under Workbench in the same way you'd play something like Minesweeper under Windows. Most games expect full control of the computer, and skip Workbench entirely.
The few games that do allow installation, like simulations (Sim City comes to mind), strategies, wargames, do allow you to install the game to the hard drive, where you can launch it from Workbench from an icon like the 'Say' application. Check the manuals for these games for how to install them; every game is different, though most games that are installable show up as a disk icon in Workbench, containing an installer icon somewhere inside.