They are beautiful areas, as others have said.
The weather in September is usually rather pleasant, but can be somewhat variable. It might be summery or it might be chilly. (This is even more true a month later in the year, when anything from snow to tee shirt weather is not entirely unexpected.) It is a good time to visit to avoid too many crowds as the kiddos are back in school and the leaf peepers have not yet come in droves. If you take the cog railway (or drive the auto road or hike) up Mt. Washington, it will be cold and windy at the top...possibly very cold and very, very windy.
The Kanc is a fabulous drive through the White Mountains with many stops worth investigating. US 302 is also a very pretty road and has a few nice stopping points at waterfalls etc. I-93 and/or US 3 (they're parallel or co-located) through Franconia Notch is a third very pretty road with some great stops, including the ghost of the Old Man of the Mountain.
I feel a bit obligated to mention that my home state of Vermont is also very lovely. :B