My darling SO gets his standard vacay next September for about five weeks.
We are rough and tumble, we don’t mind being on a budget or backpacking, in fact, we prefer it. We’re youngish (27), no kids, an ok budget (let’s say around $7K to $10K, including flight and emergencies) and like adventure and little to no planing. We will go anywhere in the world.
So, what do you suggest and why?
September has five weeks? 
Head south! It’s spring somewhere in September.
Hahaha, we don’t mind cold at all though.
For example, our current ideas are: Sailing in the Caribbean with a captain who will train us so we can get our sailing license, Switzerland and Austria to ski, or Egypt.
Varied! And adventure. That’s what we want.
Rent a campervan in New Zealand or Oz.
Wildebeest migration from the Masai Mara back to Tanzania. It’s the right time of year for it.
Venice always. September is a good time to see Italy in general. The climate’s not too hot and tourist season has calmed down by then. We did Venice, Florence, Rome and Sorrento in June. I loved it. My friend was there last September and probably had an even better time than we did. Getting around is easy. The trains are quick and easy to understand. We were able to find cheap hotels in reasonable neighborhoods.
Five weeks should allow you to see a lot.
Rent an RV and hit the road. Just go where you want. At the 2 1/2 week point, start heading back.
Spend ten days of it at the Walnut Valley Festival (commonly just called “Winfield”) in Winfield, KS. This is an acoustic music festival with dozens of acts in genres that encompass bluegrass, Irish, cowboy, folkish, old-time, etc. Past performers have included the Dixie Chicks and Alison Krauss (before they got famous), Doc Watson, and Tommy Emanuel.
Most festival-goers (there are thousands) camp on the grounds, making a temporary community with a wonderfully friendly vibe. A large percentage of attendees sing and/or play, and there’s campground music around the clock. It’s an experience you won’t find elsewhere.
The stage shows begin in earnest on Thursday and run through Sunday, the festival end. But the campground opens a week prior to that Thursday, allowing a week of camping in a more relaxed atmosphere before the bulk of the crowd arrives. The week preceding the festival and the festival proper are two different things (although they blend together) and experiencing both is the way to really appreciate this inimitable event.
Southeast Asia! Cambodia, Vietnam and Laos. Very inexpensive, good tourist infrastructure (well at least Vietnam!), friendly people and fantastic food.