Southern VT Long Trail
Okay, fine...if you insist on calling it the Vermont section of the "AT" I guess I can live with that since you will be peeling off toward evil NH after Killington.
I will refer to it as the LT up until you are heading over to NH.
I think that only you can judge if 12 -13 days is enough time. What's your hiking rate and how far do you want to go each day and how much time for re-supplies into town, etc.?
I agree with una_dogger for Manchester and Rutland as resupply towns. And that getting to Rutland may be tricky. So her advice on that and on the potential ride list from the GMC makes good sense.
I'll put a plug in for Goddard Shelter. It was great in 2004 and I was just there this past weekend and it's still great with a brand new privy! When you're on the LT, if you like being near ponds I'd also recommend a stay at Griffith Lake Tenting area instead of at Peru Peak shelter. Or consider some time at Stratton Pond Shelter. Both are really purty. Little Rock Pond would be a great place to take a rest day if the weather was right! Are you aware of the fee at many of the higher use shelters? A caretaker will often come around in the late afternoon to do a little LNT small talk and talk to you about the different kind of privies on the LT and how to properly poop and pee or not pee in each kind. For this service and taking care of said privies there is a charge of $5-6...I can't remember which. Anyway some people get all bent about this, but hey, if that's you, bring a tent and LNT stealth camp someplace off trail. Also, if the weather permits I'd recommend sleeping out under the stars on the tent platform at Cooper Lodge just below the peak of Killington. Go to sleep just after twilight and watch the stars come out one by one. Cooper Lodge itself is a bit out of wack in relation to most of our expectations of level, square, right angle kind of arrangement. So, the tent platform is better for sleeping there anyway, in my opinion.
Where the AT (now you've left the LT and are headed toward NH) crosses the White River over a car bridge there used to be a great swimming hole with a fantastic rope swing and a 30 foot jump off the bridge into a deep pool. Last summer they started construction on the old iron bridge so I don't know what the status of this fun swimming area might be. But, check it out if you like swimming holes. Also, the store that used to be right there at that road crossing is no longer open and hasn't been for a while, but it tends to fool people. The nearest store is quite a few miles from there, but I bet you could hitch it. Heck, I've picked up an AT section hiker there once, so others must do the same.
Have fun. Hope that wasn't excessive info. Late Aug/Early Sept. is a great time to go. You'll likely run into some AT through hikers at that time.
Have fun! Welcome aboard.