I don't know the Trek 1200 specifically, but it sounds like a good setup for light touring.
I also don't know Tennessee that well... But I'm sure you've got hills to climb. The reason I say that, is the gearing you mention is more suited to fast road riding (not loaded touring) and/or riding on the flats. So if you get into touring you might consider a lower range; say 48/38/28 with a 12-32. Somethin' like that. Gears are easy to change (a full set should be from about $80 on up, depending on brand and quality). It'll be a lot easier cranking up hills with even a small load.
Yeh -- BRAT sounds like a great way to introduce yourself to long rides. It depends on the level of "supported". Do they carry your full gear each day and meet you at the campground? On these rides, most people carry only a handlebar bag or small stuff sack on the rear rack. To carry a jacket, rain pants, some food, etc. You rarely see panniers on these rides. In fact, some of them look like pelotons.
The next level up is credit card touring. You could get by the same as a fully supported tour, carrying only a handlebar bag or small stuff sack. Buy everything else you need: from snacks to disposable cameras to repairs to an extra pair of shorts.
Motel touring would need at least a small set of panniers -- more clothes, picnic lunch, tools, etc.
Loaded touring is camping on wheels. If you're going out for more than a couple of weeks, you'd need a full rack/pannier setup or a trailer. Search this board for comments -- there are advocates for both. Also, there are advocates for carrying cooking equipment or not; I've decided the extra weight of a stove and fuel just aren't justified when my route passes through at least one town a day.
With panniers, at some point the weight you carry requires you have both front and rear panniers. If you try to stuff everything on the rear rack, the bike gets squirrely. General rule seems to be split the weight about 60/40 (rear/front).
You don't have to get it perfect the first time. Like any other adventure, you get better at it the more you tour. Your idea to do BRAT is a great introduction to touring. You'll learn about your capabilities, your bike, your preferences. That'll give you confidence for extended touring. The confidence edge you need, if you want to get out on the road yourself, is to learn your own bicycle maintenance (changing spokes, adjusting brakes/shifters, etc.) -- in fact, if you get new gears, swap 'em yourself.
13 mph isn't slow
Unless your name is Lance. Touring just requires that you can do 50-70 miles a day, doesn't matter if it takes 10 hours.