I had a similar problem with my Samsung Galaxy Mini GT-S5570. I found the problem was that my contacts weren't saved on my phone, they were saved on my SIM card. If you save them to your phone, then when you go into them it then gives you the option of adding additional fields such as extra numbers, e-mail & physical addresses, birthdays etc.
Try going into your contacts, and from the options choose 'Import/Export' and then 'Import from SIM card'... you will probably have some double-ups which you'll need to delete, but you should then be able to edit your contacts and enter more information under each one. You can see a SIM card symbol next to any contacts which aren't saved to your phone.