NFCT
Northoceanbeach,
The western end of the trail begins in a very popular tourist area and there will be boat traffic on the bigger of the first lakes you cross. But you will be near civilization and paddling flat water, nice for you and the dog till you get your system down.
The first 90 miles is easy paddling and there are many campsites along the rivers and lakeshore to use. Those on the Saranac Lakes require registration. The tougher paddling begins on the Saranac River between Saranac Lake village and Lake Champlain. There is whitewater, but there are portaqes around all the bad stuff, though some are long carries. Crossing Champlain is a challenge in itself in an open boat, you may have to wait for the right weather conditions. The New England and Quebec sections are mostly river travel with some upstream paddling or poling. Those sections will put you in the most remote areas.
The people at the NCFT are very helpful and their maps were made by the people who laid out and developed the trail, so they are accurate and detailed. Contact the NCFT, join the organization, and make use of a wonderful resource.
Bill
not only do you have to carry
all your own gear in the NFCT but much of the time you have to carry your boat too—lots of portages.
you might try a longer trip on
the Missouri—kind of Lewis and Clark like only in reverse(I always find it more relaxing to paddle with the current and wind.) The NFCT does involve a LOT of portages—rather like hiking the Appalachin Trail carrying a boat. 
I would urge you to follow some of the advice above and do a shorter trip first—say between 5 and 14 days just to acclimate yourself to canoe tripping and see if your fantasy is at all close to reality.
I believe the eastern end of the trail
goes down the St.John River and ends in the town of Fort Kent. During May and earl June is the only time the Upper St John has plenty of water and there are ClassIII rapids at Big and Black Rapids near the end of the trip(note these area not present at lower water leveles latter in the summer but then you may have to drag your canoe through) At their most runnable, those rips are not for beginners.
I’ve been thinking of the paddling
the NFCT myself and have done a little research. From what I read, it seems like using a canoe cart for the portages would be OK. With a cart the portages wouldn’t be too bad even if long. I found that it is easy to get helpful firsthand info on the trip. It wouldn’t be too hard of a trip to plan because of all the maps and other resources out there. I’d bet you can contact local paddlers/clubs for the various parts of the trip that would be willing to help you out. I personally would feel relatively safe doing it solo.
Carts on carries
Some carries use roadways and are very cart friendly. Some others can be carted for the majority of their length but have obstacles that require lifting the canoe and cart to pass over.
Two people with a tandem can do these short lifts easily, a solo paddler with a loaded canoe cannot. The best carts for these carries are carts that do not have an axle connecting the wheels, and have large diameter bicycle type tires, both for clearance over rocks and stumps.
The original poster is taking a dog, and the dog could be equipped with saddlebags to help transport some gear. I have done the first 90 miles of this trail over 15 times now and am very familiar with the carries; I have done them with everything from a solo canoe to a 20 foot Grumman aluminum war canoe. The stone steps at Raquette Falls can not be carted solo, and the stone water bars trap most small cart wheels. Have seen a lot of kayak carts destroyed on this one carry, and many others pulled from the end of the kayak. A good yoke and pads is essential on the NFCT even if you are taking a cart. Some places are much easier to carry overhead than wheel.
Bill
How safe is your gear when you
have to make more than one carry?
Depends
The portage location, what you leave unattended, how long it takes to make a round trip, all have an effect on whether someone makes off with your gear. A portage with road access is the biggest risk. The farther from an urban population, the safer your stuff. If the people going by have the same stuff, why take yours and have more to carry. If the round trip takes an hour or more, someone could get the idea you forgot the stuff.
You don’t have to carry each trip from end to end on the longer portages. Just carry till you start to feel the load, put it aside the trail and return for another load. Breaking up a long carry gives your shoulders a rest, keeps your group closer together, and leaves your stuff unattended for shorter intervals.
Bill
depends on if you lose it
in the boreal forest bushwhacking is a necessity at times and there is a real possiblity of not being able to find your gear again..
of course a compass helps and flagging tape.
Sometimes people coming from the other way are nice and carry over that second load!
You might want to investigate FSCPT
You may want to look into the Florida Circumnatigation Saltwater Paddling Trail. This encompasses the entire coast of Florida from Pensacola to Jacksonville via Key Largo. You can break this into sections and experience both wilderness and urban paddling. Lots of camping opportunities, particularly along the Gulf Coast. There is plenty of access and if you need to bail out at any time, you can do so. For a paddler with limited experience, the section from Carrabelle Beach to St Marks might be a good choice. To extend your trip you could also begin at many points to the west of Carrabelle all the way to Pensacola. Some of the areas might test your skills however.
For a more urban trip but still with camping opportunities, you could put in near Crystal River and head south toward Naples. South of Naples you get into the Everglades and this section requires considerable navigation and wilderness skills.
After making the turn north, you get into the relatively urban sections from Miami to Melbourne. Less camping, more hotels. Its fun, but powerboats may be a concern. From Melbourne north to the end of the trail can be quite scenic and you are somewhat cooled be Atlantic waters.
For all of this, assess your skill, use your head, and take plenty of drinking water even though opportunities for resupply are frequent.
Check this out at http://www.dep.state.fl.us/gwt/paddling/saltwater.htm
Consider the Bowron Lakes circuit
… in Bowron Lakes provincial park in northern BC, Canada. 110 km circuit through lakes and a bit of river. About 10-12 km of portage. 50 campsites.
Here’s a short impressionistic film clip of our trip a couple of years ago:
http://web.mac.com/guillermobarron/iWeb/Guillermo%20Barron/Bowron%20Lakes.html
second
Nothing wrong with the Miss. I live near it and frequently canoe it. NE MO. Good scenery. Pollution? You don’t want to drink it, but there are walleyes and northern pike in it where I live and they don’t like water that is nasty. Sure, there are warnings not to eat more then a couple of pounds of bottom feeders (catfish a week). I took a class on the Mississippi two years ago that explained all about the river (and we talked to a lot of people who “use” the river) and it is OK. Just enjoy and you will be fine.
what datongdave eluded to…
If you are still a beginner, and take on big, moving water....IMHO, you're going about it(paddling) in the wrong manner... You'd enjoy it much more if you'd learn it by improving your skills by going from "easier" to "tougher" water....usually starting with clean, cool-water ponds.
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