East coast backpacking/camping

My absolute favorite Northeast backpacking route is the Long Trail in Vermont. It's somehow the perfect mixture of rugged, beautiful, and rolling. I'd highly recommend it. Unless you're in ultra-runner shape, you won't be able to hike it in a week.

The Northville-Placid trail in the Adirondacks is doable in a week if you're in solid backpacking shape. You could also break it up and do it in shorter sections.

Then there are the "Ultra Eight," eight of the most difficult day-hike trails in the nation. Most/all are usually broken up into multiple days, and all would make an excellent weekend backpacking trip. They consist of:

The Great Range Traverse (Adirondacks, NY)
The Devils Path (Catskills, NY)
Mahoosuc Trail (NH)
Saranac Ultra 6 (NY)
Presidential Traverse (NH)
The Pemi Loop (NH)
Taconic Crest Trail (multiple states)
The Cranberry Lake 50 (Adirondacks, NY)
I'd add the Escarpment Trail in the Catskills of NY to the list as well

Other options to consider are hiking sections of the Appalachian Trail, the Long Path in NY, or putting together your own adventure.

In general NY and VT are simple - just go and hike. In NH, specifically the White Mountains, the AMC regulates things. Backpacking can require reserving space in one of their huts, at an extremely high cost. There are also some backpacking campgrounds, also run by the AMC, and which also cost money. In Maine, specifically anything in Katahdin, permits and reservations can be necessary.

The high peaks in the ADK's- literally anywhere Lake George North