I've never had an Amiga, unfortunately. But I am familiar with Commodore 64. What you will need:
-The Computer itself (duh) - The keyboard is built-into it.
-Power Chord
-A T.V. Hook-Up or a Monitor (T.V. hook-up is cheaper)
-A Floppy Drive - Technically you don't need this, it does have a cartridge slot built-in, but the large majority of C64's games and programs came on floppy.
-A basic understanding of BASIC computer language, or you'll have a helluva time getting anything to load up.
-And a joystick if you plan on gaming. An Atari 2600 joystick will work fine. Unfortunately, unlike the Atari 2600, a Sega Genesis controller will not work. I've heard it can short your system out.
If you aren't worried about getting official copies of games, the good news is that games were notoriously easy to copy onto blank floppies on C64. So most games can be obtained quite easily.
Don't be surprised by long-load-times. I daresay that the C64 has the longest load-times of any system, ever. I swear I could set a game to load, go make a sandwich, come back, and it's still loading. Except for cartridges, those are instant. Speaking of cartridges, Epyx makes a "Fast-Load" cartridge that cuts down on floppy load-times somewhat.
There was also a cassette drive. I know little about it, as I never had one. I still played all the game I wanted to play on my C64 without it