The engineers who deal with ship design are called naval architects. Training as a naval architect involves not only learning about engineering ship structures, especially those made out of steel, aluminum, or composites, but you also get the basic undergrad engineering course material (at least in US schools) which is common to many other engineering degrees, like mechanical engineering, for example.
Not only are naval architects trained in structural engineering, but since a ship floats, there is extensive training in ship stability, to make sure the vessel floats and floats upright, and the motions of vessels in waves. You are also exposed to vessel design, the tools with which you can estimate the size of the vessel required to carry its cargo and serve its intended route of operation, and how to lay out the internal spaces of a ship. For powered vessels, you are also taught how to estimate the power required to attain the vessel's service speed.
In the United States, there are a handful of schools which grant degrees in naval architecture. Typically, the undergraduate degree in N.A. is a four year program resulting in a B.S. degree. The U. of Michigan at Ann Arbor and the U. of California at Berkeley are the largest school offering such degrees, both at the undergrad and grad level. MIT in Cambridge, MA offers both undergrad and graduate degrees in naval architecture. All three schools are quite well known in this field and have trained many prominent N.A.s over the years.
Michigan has a school of Naval Architecture and Marine Engineering. UC Berkeley and MIT have departments of Ocean Engineering, which share facilities with the Mechanical Engineering department:
http://www.me.berkeley.edu/Grad/Areas/ME_Main_Frame_Ocean_Eng.htm
http://name.engin.umich.edu/
http://oe.mit.edu/
There is one school, Webb Institute, located on Long Island, which offers one program, leading to a dual degree of B.S. in Naval Architecture and Marine Engineering. Marine Engineering focuses on the design of propulsion and auxiliary machinery installed in ships. Webb has one program, is housed on its own campus, which is a former estate on Long Island Sound, and the student body is small: the entering class is about 25 and the entire student body is never more than about 70 students total. Although only an undergrad degree is granted, graduates of Webb usually have no problem getting into the graduate programs at MIT or Michigan if they chose.
Webb also requires that its students also be exposed to work in the marine industry all four years of the program, with the first year working in a shipyard, the second year sailing onboard a ship, and the final two years working in a design or engineering office. One of the attractions of Webb is that on acceptance, you are basically awarded a four-year scholarship, since the tuition is free, and you are expected to pick up the modest room and board charge and buy your textbooks. Students live on campus in the estate. (disclaimer and plug: I am a graduate of Webb)
http://www.webb.edu/
If you wish to study abroad, there are several good schools in the UK, France, Germany, and the Netherlands.
One word of caution: don't go into these programs unless you like to study and have a rather good preparation in math and science at the high school level. Because of the amount of material covered in four years is quite large, you will be taking about 20 hours a semester every semester you are in school. These are core subjects, and there is not much time left for electives or extracurricular activity.
Shipbuilding is a global business nowadays. One need not live near a shipyard in order to be a naval architect. Drawings and other technical data needed to build a ship can be prepared in one country and sent to another by the miracle of the internet.
Good Luck!