There are pros and cons for each age of bird.
Chicks: Chicks are nice because they are inexpensive, tame easily, and are usually readily available in many different breeds/varieties. They are less likely to carry disease, too. And of course, they're adorable.
Watching chicks grow from tiny fluffballs to mature, egg-laying chickens is very rewarding, as well as educational. Disadvantages include: more labor requirements, a longer wait for eggs/meat, and fragility of the chicks.
Adult birds: Adult birds have a few advantages. Firstly, they take a lot less care than chicks do--no special brooder requirements or nurturing. Eggs may already be being produced, or are very close, which is a benefit if you can't wait for the eggs or have a commercial-type operation. Adults birds are stronger than chicks, as well. Some disadvantages include: greater likelihood of them carrying disease, and a flighty nature. Adult birds don't tame very easily.
Personally, I prefer chicks. However, the "best" age of bird does vary for each person.