You can grow organic vegetables in container pots. This is a easy method. It also allows you to avoid soil grown diseases.
To get started look for organic soil from your local store. Then fill your pots with the soil. Start with vegetables like carrots, lettuce, and tomatos. You can also grow a number of fruit or herbs as well. This will let you get into the swing of things before you more on to more complicated vegetables.
The best way to do this is to grow what you really enjoy eating the most. Start with that and you will reap the most benefit from it.