As an Olde Phogey who's been building plastic models far longer than airbrushes have been in the hobby, I'm a big believer in brush painting. These days I do have an airbrush, but I rarely if ever use it for ship models.
Each modeler develops his or her own tastes and skills; what works well for one may seem awkward to somebody else. My own preference is for acrylic paint. My favorite brand is Poly-Scale; I haven't had as much luck with Testor's Acryl, though it's probably more widely available. I like the way Poly-Scale goes on with a brush, and the range of colors is tremendous. My only complaint is that, for some purposes, it dries too fast.
Many veteran modelers swear by sable-hair brushes, and I agree that they're excellent. I also, however, am a big fan of synthetic-bristle brushes. A trip to an art supply store will reveal a huge range of them in all shapes and sizes. I've gotten equally satisfactory results from the white and gold bristles - and they're a good bit cheaper than the sable ones. Take a look at the variety of shapes available. If I were picking out a set of brushes from scratch for doing 1/700 ships, I'd probably get a 1/4" flat (for the hull), a smaller flat (for the decks), a #3 round with a pointed tip (for the superstructure and larger detail parts), and a #0000 or #00000 round (for the smallest details).
Acrylic paints take a little practice. Thin them (preferably with distilled water, but for brushing tap water probably will work fine) to about the consistancy of thick cream for starters. (Poly-Scale has about that consistancy right out of the jar.) You want to lay on a thin coat of fairly thick paint. Unless there's a big contrast between the base color and that of the paint (e.g., you're trying to put bright yellow paint on black plastic), one coat probably will be enough. With a little practice, you can lay the paint on with virtually no visible brushstrokes. And if you screw up, as long as you recognize the problem within an hour or so a rinse in fresh water will get you back to square one.
Get in the habit of covering every square milimeter of the model with paint - regardless of the color of the plastic. Unpainted plastic has a slightly translucent, waxy sheen that's at odds with the scale effect.
Don't get scared off by people who prescribe specific colors - and don't be intimidated by the weird color names on the bottles. Do as much research as you're comfortable with, and pick colors that look right to you. The fact that it says "Italian Air Force Yellow-Brown," or "RLM 63," or "Conrail Green" doesn't bar it from use on a ship model.
Above all else, keep in mind that it's a hobby and the primary purpose of a hobby is to have fun. I'm afraid newcomers sometimes get the impression that scale modeling is the exclusive domain of experts - experts with deep wallets. If the finished model looks good to you, you learned something from building it, and you had a good time in the process, that's what matters. End of sermon. Good luck.