Jiangsu

The main rail corridor south of the Yangtze, Nanjing-Shanghai, running through Jiangsu's biggest cities, now has three more or less parallel railway lines: the old ("conventional") Beijing-Nanjing-Shanghai line (lowest fares, but intra-province tickets are hard to come by; mostly used for taking trains to other parts of China), the high-speed Beijing-Nanjing-Shanghai line (mostly useful for taking high-speed trains to other parts of China), and the "intercity" Nanjing-Shanghai line (frequent service between region's cities). Many cities in this corridor have separate stations for each line, so always make sure you know where your train is to be boarded. Nanjing has two railway stations (Nanjing and Nanjing South) as well.

North of the Yangtze, the Nanjing-Yangzhou-Nantong line has frequent high-speed service, along with some conventional service as well.

There is no passenger rail connection across the Yangtze downstream of Nanjing (cargo trains use a ferry), so you can't take a train from e.g. Yangzhou or Nantong to Shanghai. However, there are talks about building a railway bridge around Nantong.

Xuzhou, in the far north-west of Jiangsu, is a major railway hub. The rest of northern Jiangsu has a number or rail lines as well; service between most of its major cities and Nanjing is available, but is less frequent than on the lines along the Yangtze.

The roads are good so you could consider renting a car and driver, or even driving yourself. See Driving in China.

Outside of small historic city centers, Jiangsu's cities have fairly good conditions for bicycling, with wide street provided with separate lanes for two-wheeled vehicles (which, these days, are mostly electric motorcycles rather than pedal bicycles).

Traveling by bicycle between cities is more pleasant in wealthier Jiangsu than in many other parts of China (and Asia in general) as well. In many other parts of the country it is crossing a city's outskirts which is the least pleasant part of a bike trip: central cities have decently maintained streets with bike lanes, countryside has fairly light traffic, but cities are surrounded by belts of light industrial land use (rebar vendors, cement factories, tire shops, recyclers, etc), where roads are often broken and dusty, while traffic is heavy and chaotic. In Jiangsu, however, it is often possible to leave a city via a wide boulevard with stately buildings and parkland around it, such as Nanjing's Xianlin Road (仙林大道).

Most of the bridges across the Yangtze only carry expressway traffic, and can't be used by bicycles. To cross the river with a bike, you can use Nanjing's First Bridge (the city's oldest bridge, NW of downtown), or a ferry (such as the one between Yangzhou and Zhenjiang).

Many cities (e.g. Nanjing, Zhenjiang, Yangzhou) have bicycle rental programs, with self-service bicycle stands around the city. However, one probably has to be a local resident with all the accoutrements of residence (a local ID card, a Chinese cell phone, a Chinese credit card...) to make us of it; see if your local friends can help you.