All the above is excellent advice. Remember, though, that Borden makes at least two types of wood glue with the name "Elmer's."
The old-fashioned, pure white stuff is actually a very good adhesive for most purposes. I use it a great deal for securing rigging knots and for detail work. One of its biggest virtues is that it shrinks a little as it dries; another is that it dries almost transparent. It also can be softened up later with the application of a little water.
"Elmer's Carpenter's Glue" is quite a bit different. It's a yellowish cream in color; I think it's virtually the same stuff as Franklin Titebond. Both of them are superb adhesives for wood-to-wood joints. They're stronger than traditional white glue, dry faster, don't gum up when sanded or planed, and generally are the preferred adhesive for structurally important wood-to-wood joints these days. Just be careful to keep the glue where you want it. If it gets on a piece of wood (especially wood you intend to give a clear finish) where you don't want it, wipe it off with a damp cloth before it dries. Otherwise it will play havoc with the finish.
Cyano-acrylate adhesives, otherwise known as superglues, have been knocking around the hobby industry for roughly thirty years now. I tend not to use CA for wood-to-wood joints, but it has all sorts of other uses - especially in joining dissimilar materials (e.g., metal-to-wood or plastic-to-wood). I've heard stories of CA glues failing over time, and many traditionalists don't trust them. All I can say about that subject is that I've got a couple of ship models that I finished in 1979 and 1984, respectively, on which I used quite a bit of CA glue. The CA brands available when I built that first model probably were inferior to those available now. (Any other olde tymers remember Aron Alpha, which was the first one I encountered?) But not a single glue joint on either of those models shows any sign of coming loose or weakening.
I'll take the liberty of offering a suggestion regarding the balsa wood parts in the old kits. Unless you're primarily interested in the kits as artifacts in the history of model building, throw the balsa out. Balsa is, by nearly universal agreement among experienced ship modelers, the worst wood in the world for scale ship modeling. It's fuzzy, too soft to work effectively, coarse in grain, and almost impossible to finish decently. (Ok, I've seen some nice, smooth finishes on balsa, but they took a ridiculously long time to achieve.) Balsa became the cheapest and most widely available hobby wood for one reason: it's light. That fact is important to builders of flying model airplanes, but not to scale ship modelers.
The subject of woods for ship modeling is a complex one on which every modeler has a different opinion. An easy, practical approach for a beginner, though, would be to visit a decent hobby shop (local or online) and buy enough basswood to replace all the balsa. Basswood is nice stuff to work with - a good deal harder than balsa, very reasonably priced, and widely available. A good hobby shop will carry basswood in as many different sizes and shapes as balsa.
There's a widespread misconception that balsa is easy to work with because it's soft. Please believe me: that just isn't true. To get good results from balsa wood requires a real master woodworker - though such an individual probably wouldn't have anything to do with it. Harder woods are far more patient with and forgiving of the beginner. Just be sure to keep your tools sharp. If you're using X-acto type blades, lay in a supply of spares and throw them away as soon as they get dull.
Hope this helps. Good luck.