Well, this can get kind of complicated - depending, of course, on how deeply you want to study the subject. Let's start with the question of line sizes.
In a real frigate of that period the number of sizes of rope ran into the dozens. Few modelers try to reproduce every one of them - especially on the relatively small scale of 1/96. But the more different sizes of thread you use, the better the model will look. That's one of the easiest improvements one can make to a kit - and thread doesn't cost much.
There's a detailed table of rigging sizes in the appendix to James Lees's The Masting and Rigging of English Ships of War, 1625-1860. (It's reasonable to assume that an American frigate's rigging would be pretty similar to a British one's.) Mr. Lees says the size of the lower shrouds was 0.6 times that of the corresponding stay, which in turn was half the diameter of the mast.
Let's take the main shrouds as an example. The Constitution's spar dimensions changed several times during her career, but the mainmast seems generally to have been about 3' in diameter. (A set of her spar dimensions that I have in front of me says it got as small as 2'8" for a while, but let's use 3' to keep things simple.) Half the diameter of the mast would be 18". So the mainstay would be made of 18" rope. Multiply 18" by 0.6 and you get 10.8". (I keep a calculator that works in feet, inches, and fractions handy at the workbench for occasions just like this.) So the main shrouds would be made of 10.8" rope.
Unless it says something to the contrary, it's safe to assume that any source describing rope sizes is referring to the circumference, not the diameter. In picking thread for a model's rigging, the diameter is far more useful. To find the diameter, divide the circumference by pi, which is 3.1416. 10.8" divided by 3.1416 is 3.4377387". So let's say the main lower shrouds of the Constitution were three and a half inches in diameter. (That's pretty hefty rope.)
Just one more step. The model is on 1/96 scale, so to get the diameter of the thread you should use for the main lower shrouds, divide the full-size diameter by 96. Divide three and a half inches by 96 and you get .0364583". That, ideally, is the diameter of the thread you're looking for.
Two golden rules of ship model rigging. 1. If in doubt as to color, err on the dark side. 2. If in doubt as to size, err on the small side. Thread that's .035" in diameter will be fine, and .030" probably will be more than good enough for most observers.
To make things simpler - use 3 instead of 3.1416, and 100 instead of 96. If you're working on a 1/96 scale model and you know the circumference of the actual rope you're trying to reproduce, divide that figure by 300 and you'll be close enough for most purposes. (If you use that figure instead of the ones above, your thread works out to be .036" instead of .0364583. Close enough.)
Ratlines, as you've noted, were much thinner than shrouds. (The shroud has to hold up the mast, and transmit the force of the sails to the hull, thereby pulling the ship through the water. The ratline just has to support the weight of a human being.) Mr. Lees says ratlines were generally made of inch-and-a-half rope. One and a half inches divided by 300 equals .005" That's the size thread you want, ideally, for your ratlines. In practice, if you use the smallest sewing thread you can find, it won't be far off.
What sizes of thread are available at your local craft store I have no idea. But why not order some thread online? Model Expo (www.modelexpoonline.com) and Bluejacket (www.bluejacketinc.com) both sell high-quality rigging line in a wide variety of sizes. The difference in price between high-quality line and the stuff at the craft store won't amount to much - certainly not in comparison to the time you're going to be putting into the rigging of that model.
Some modelers may disagree with me, but I don't recommend CA glue for most rigging applications. If I were you I'd thread the ratlines through all the shrouds (including the first and last ones) and secure the ends (only the ends) of each ratline with a tiny drop of white glue. In such small quantities it only takes a few minutes to dry (though I'd wait a couple of hours to trim the ends of the ratlines, just to be safe), and it doesn't stiffen the thread like CA does. White glue also doesn't have the ugly, shiny appearance that CA has. (If the white glue does show when it's dry, you can touch it up with some appropriate flat paint. Don't use acrylic for that job; it will soften the glue.)
Somebody else will have to offer suggestions about plastic sails. I personally just can't live with them - but I guess that's a matter of personal taste.
Hope that helps a little. Good luck.