Definitely Annapurna (northern cicruit)
In September, I would not take the risk hiking the Everest region due to monsoon season (although one might be lucky if this year’s monsoon has an early retreat – it’s a hit or miss).
The northern section of the Annapurna circuit is largely on a rain shadow. You might have cloudy days but chances of rain are far less the in the Everest region. The advantage of the Northern Annapurna circuit is the variety of climate zones one traverses (check out my profile for pictures).
Both the Annapurna and Everest region can be hiked fully independent, without a guide. Plenty of people do it. The northern Annapurna circuit implies getting a bus to the trailhead, currently Besisahar (there is normally a daily direct bus from Kathamndu) and flying out from Jomsom (only flights to Phokara, which implies taking a connecting bus or flight back to Kathmandu). One can optionally skip the first and second day by taking a shared jeep further on from the Besisahar.
If it’s your first time in Nepal, it might be useful to get assistance from a trusted local travel agency to get the bus- and plane ticket (PM me if you want a recommendation, although most hotels will also assist you – just insist you want to trek independently, without a guide as they might push you to take one as it generates more revenue).
Once on the trail, trekking is dead easy. Plenty of accommodation & restaurants in virtually every village you’ll come along – no reservations needed whatsoever, just pick the teahouse you like. English is widely spoken. Prices are more or less the same everywhere. If you don’t splash out, you’ll get by with 25 usd/day (excluding transporation, Annapurna entrance fee and tims-permit). No need for tent, stove or food (but a sleeping bag remains necessary).
Also very difficult to get lost as you follow the main valley.
A good guidebook is paramount when trekking independently. I would nowadays recommend both the LP trekking guide for Nepal and the Cicerone guidebook on the Annapurna region. Contains all necessary information. The best guidebook used to be the Trailblazer but they are not available anymore. Both LP & Cicerone guidebook can be purchased as ebooks
Also plenty of information on the LP forums, that can supplement your guidebook information.
If you have cash to spare, you could ditch the Annapurna idea and go to Mustang, completely in a rain shadow (not allowed as an independent trek though – you’ll have to take a guide, get a 500usd permit and travel in a group of at least 2). Mustang can be booked locally through a trusted travel agency and can nowadays be done as a teahouse trek. Mustang and the Northern Annapurna circuit can also be combined.
If you feel uncomfortable going independently or booking guided trips locally, check out Project Himalaya. Relatively expensive but top notch organization and plenty of options. One can’t go wrong with Jamie McGuinness as a guide – just google his name.
PS: if you haven’t booked your Nepal flight, one could consider Ladakh over Nepal for September. Ladakh completely situated in a rain shadow, offering superb hiking conditions during the summer months.