Pick and chose that which appeals to you. The first herb that comes to most people's mind is arguably Mint. All mints, from the licorice, chocolate and apple flavored kinds to the candy cane peppermint are aggressive plants. So aggressive in fact that once planted you may never be rid of it and it will go everywhere and possibly choke out other plants. Believe me on this one.
"But I like mint" you may be saying. Truly, no one likes mint that much. If you want to grow mint despite all the warnings, do so in a large, clay pot next to the herb garden to at least attempt to curb its wandering ways.
The Cuban type grows more like a succulent so it doesn't dry well.(It freezes fine). Snip a piece off and you can root it in water very easily. That's how I grow new plants to give away. I gave one Cuban oregano plant to a friend and she told me it actually got her husband cooking for the first time. Anything I can do to help the busy working woman!
Lemon balm is another perennial herb which makes a wonderful, calming tea. It grows to be around 2 feet tall and if left to go to seed can produce many offspring. This can be dried or frozen for later use.
Chives is one of those plants I love more for the flowers which I pack into jars and fill with plain white vinegar to make pretty blushing vinegar with a slight oniony taste. This is great on salads. Woe to those who allow chives to go to seed however!......read more.