go and get yourself a copy of the Appalachian Trail Thru-Hiker's Companion. it's pretty much tells you where shelters/water sources/access roads (and the distance to and from each one) and lots of other things. it will save your life!
250 miles.. well, it really depends on how fast and long you walk and of course your pack weight. mine was about 25lbs - 35lbs with a weeks food and as much water as i could carry. i've been backpacking since i was about 8 or 9, so i'm kinda used to knowing what to expect.
for most people, first day or two- i'd say between 5 and 8 miles p/day (you should be training before you go and fatten up, too). after a few days your muscles will get more used to being so active and you can get up to 12-15 miles a day.
you're not thru-hiking so you don't really have to rush it. i had to hike at least 15m a day just to my destinations in time.
shelters are set up at about half a day to a day's hike apart(again, depending on how fast you walk) and most have some kind of water source and some have bear cables to hang your food.
hitching into town is soooo easy around the trail and people will sometimes give you food or pay for your meal or something. those people are called "trail angels". they are rad-ass.
also a lot of the convenient stores won't accept EBT. i found this out the hard way. bring some moneys or a debit card.
depending on your pack-weight, you could do it in a month... give or take.