The answers to these questions, of course, depend largely on personal taste.
I'm assuming, for the moment, that the subject in which you're interested is twentieth-century warships. (Sailiing ship models are a quite different game.) The most popular scale in the model warship world at the moment - at least in terms of kit availability - probably is 1/700. The advantages of that scale are significant. If you count the resin "cottage industry" kits, more 1/700 kits are available than a person could build in a lifetime. Even the ranges of the mainstream, injection-molded kit manufacturers are enormous, with the majority of major navies' ships since 1939 well represented. (Ok, there are significant gaps, but virtually all the really famous ships are there.) And the resin manufacturers offer an incredible variety of ships - more than most of us have heard of. There is, for example, a Russian firm called Combrig that offers most of the steam-powered ships the Russian and Soviet navies ever built. And Combrig prices, due to the currency situation, are very reasonable.
1/700 kits are relatively cheap. You can buy quite a few evenings' worth of model building for $10 to $20 (though some of the larger, more exotic resin kits sell for $50, $100, and more). They can be finished relatively quickly; an out-of-the-box destroyer will take a couple of evenings (though the time starts multiplying fast if you add detail parts - see below). And a relatively small cabinet can hold a pretty large fleet.
The big drawback to 1/700 scale, of course, is that it's mighty small for detailing. Several aftermarket companies, however, offer photo-etched detailing sets that are incredible in their detail. (Example: a radar screen that's less than 1/4" in diameter and consists of eight pieces.) Just a few years ago it was taken for granted that 1/700 models couldn't really be taken seriously in terms of detail. That's changed. The amount of detail some enthusiasts cram into those little ships is hard to believe.
The other popular scale at the moment is 1/350. The ranges of kits from the major manufacturers in that scale isn't nearly as large - though it's growing all the time. There's also a big range of 1/350 resin kits from the cottage industry folks. Here the price of resin comes into play. Resin kits on 1/350 scale tend to be quite expensive - the big ships go up into the hundreds of dollars.
Many of the differences between the scales are self-explanatory. A completed 1/350 battleship or carrier will be a major feature of a room. The number of pieces in the typical 1/350 kit is considerably larger than that in its 1/700 counterpart, so the 1/350 model probably will take longer to build. The job of detailing it will, on the other hand, be considerably easier (though 1/350, by the standards of aircraft and armor modelers, is still pretty daggone small). The range of photo-etched and resin detail parts for 1/350 kits is not, perhaps, quite as broad, but the quality of those photo-etched sheets frequently is downright mind-boggling. The British firm White Ensign models, for instance, sells several sets to detail the Trumpeter 1/350 U.S.S. Hornet. The sets cover parts for the guns, catwalks, hangar deck framework, and (I still have trouble believing this one)
the interiors of the aircraft. To build a Hornet that incorporated all those sets would take months - and cost several hundred dollars.
The quality of the warship kits being released nowadays is, generally speaking, quite high. The boom in 1/700 scale warships got started about 25 years ago when four Japanese manufacturers, Tamiya, Hasegawa, Fujimi, and Aoshima, formed a consortium to produce models of WWII Japanese vessels. Initially their kits varied quite a bit in quality; I suspect most enthusiasts would rank them in the order in which I just listed them, with the weakest of the Aoshima kits being REALLY weak. Since then several other firms have gotten on the bandwagon. Skywave kits - especially the most recent ones - are outstanding. Recently Dragon (most of whose 1/700 kits are reboxed Skywave products) has released three excellent kits on its own: the U.S.S. Arizona and the German battleships Bismarck and Tirpitz. Trumpeter has also gotten into the 1/700 game. I haven't bought any Trumpeter kits, but they have a fine reputation.
The list of mainstream, injection-molded kits on 1/350 scale is still pretty short. Tamiya started the modest boom in that scale about 25 years ago with a small series of battleships: the U.S. Iowa class, the Japanese Yamato class, and the British King George V class. Shortly thereafter the same company did a 1/350 Enterprise (the modern one). Those kits still look pretty good, if not quite up to today's standard. More recently, Tamiya introduced a 1/350 destroyer, the U.S.S. Fletcher, that frequently gets recommended to newcomers. It's a nicely detailed kit of an important ship, and the relatively small size means it's reasonably priced and not too time-consuming.
Trumpeter has plunged into that region in a big way recently, with American carriers of the Yorktown, Essex, and Nimitz classes and a Liberty ship. That firm has announced several additional kits that have enthusiasts' mouths watering: H.M.S. Hood and the U.S.S. North Carolina, Lexington, and Saratoga. The Czech company ICM has released a couple of German battleships from World War I (a grievously neglected period, in my opinion), but they seem to be a little hard to find at the moment.
Which scale to pick? Think about your own interests and aptitude. A big collection of 1/700 kits can be built up fairly quickly for a reasonable amount of money - if you aren't interested in going the "extra detail" route with the photo-etched aftermarket parts. Any 1/350 kit will be a big project unto itself. Another suggestion I always offer when people ask me such questions is - consider your physiology. If you don't have excellent close-up vision, working in 1/700 scale will be a frustrating experience. (I'm lucky in that regard. I have trouble recognizing people from across a room without my glasses, but I can build models with no magnification.) And some people are better working with their fingers, while others do better with their wrists and elbows.
All this is personal opinion, of course. I suspect this post will attract quite a few comments. On one thing, though, I suspect all of us can agree: this is a great time to break into ship modeling.