There are currently no direct flights from Nanjing airport to Shanghai, but to Beijing, Hong Kong, and major cities abroad you will find plenty of airlines. The airport is best reached either by taxi or airport bus. The airport is in Lukou and taking a taxi from downtown will cost 100RMB or more (there's a 20RMB toll about 3km from the airport). Drive time is around 30 minutes. There are very nice coaches which will take you to about four different locations in Nanjing; fare is 25RMB.
There are numerous daily departures to Shanghai, about two to three hours away. Keep in mind that soft seat (1st class) at 79 RMB is not that much more expensive than hard seat (2nd class) at 49 RMB and offers you much better equipment, comparable with a business class flight. The train cars are great as well, and much more spacious than those of an airplane.
There is a modern highway system between Shanghai and Nanjing, which will allow you to travel quite quickly from city to city. Keep in mind of traffic in morning and evening. If you're just one person, it's much cheaper to travel by train, if you're larger group, car gets cheaper. But keep in mind that you need to be a very experienced driver to participate in Chinese traffic, better use trains/buses between the cities and taxis in the cities, unless you're really on for a challenge.
It's not really possible to travel by boat to there, even though Nanjing is situated on the Yangtse river. The river is mostly used for transport of goods, not for persons. Better to use the railway.
Taxis start at 8RMB and there are many buses available, too. Most of the streets have a gated section for bicyclists so riders are relatively safe from passing vehicles.