this has been mentioned, but i'll type more anyways. .. ,p
1. you need to get memory card for gamecubes. There are a few choices, but i'd recommend get the 251 block nintendo brand (usually it's black). The reasons for that are: a). 251 block memory has more space, and it costs about the same with the smaller 56block memory card.
2. for most games, you have to wait until you reach a certain "save point" place before you can save the games. In the case of metroid prime, you will encounter save spots scattered around the game, where you can save and recharge your energy/ammo/etc. When you encounter these save spots, that's where you save.
3. When saving (and loading), make sure you have your memory card plugged in ,)
4. Each games has instruction manuals that give info on how or when you can save games. Look at the manual and find out
5. If you want to delete old save games (for example, you run out of space, etc), you can start your gamecube without any games in it, and go to the memory manager section. They will have the list of save games you have saved, and things you can do with them (copy, move, delete, etc).
happy gaming -)