Good old "Parsley-buttered potatoes" would be a great side for the Tarragon Chicken. As for dessert? I don't know your eaters' preferences, but chocolate is rarely turned down. Chocolate mousse is SO easy to make and always looks like you really worked at it.

Tarragon is a rather assertive herb in it's own right, so I wouldn't stray too far from the parsely path. Chives would also work with it for potatoes.

Ditto on the thyme or dill. For dessert, you could try an ile flottante or pavlova.

I would go with something like parsley, thyme or dill. For dessert - something with citrus. I just made cupcakes flavored with orange, frosted with a buttery orange frosting... pretty yummy.