If you choose to undergo the necessary procedures for hiring a car, or even easier, borrowing a friends, then where to go?

The best places for day trips around Shanghai are Hangzhou, Suzhou, Yangcheng Lake (in Suzhou) and Zhouzhuang.

Yangcheng lake is a very popular day-trip for the locals, as the lake is famous for its delicious crabs. Go in October to December for the best crabs. To drive there, you just need to get onto the Shanghai-Nanjing expressway -- the A11 -- which you can find by getting on the Yan'an elevated road and heading towards Hongqiao Airport. You can just stay on the A11 and exit at the Kunshan exit. (You can drive to Nanjing in about 4 or 5 horus by simply staying on the Shanghai-Nanjing A11, somewhat unsurprisingly.) That's a rough driving guide -- this is a basic map, and the Shanghai Tourist Map's that you can buy off the street from the phone-card sellers should be better, and sufficient.

Here are three good tour-guide sites on Suzhou, and also Hangzhou. These two cover Zhouzhuang as well.

Those sites will provide a wealth of information, but basically: If you go to Suzhou, then the main thing on your list is going to be simply seeing an old Chinese garden. There are several of these and each has little historical specialty, so any one you pick will serve its purpose. You can hop on the back of a bicycle/rickshaw and get your fat ass carried around the tourist sites for less than 50RMB (although feel free to tip, of course, though there is no absolute necessity). You can also buy a wedding dress in Suzhou, as the vast majority of Shanghainese brides do, as Suzhou is famous for cheap, good quality dresses. This is just one firm.

Hangzhou is at least an over-nighter, and if you pick sunny weather then you'll want to stay around West Lake as long as possible. Try to go slightly further out in a taxi (about 5 or 10 minutes in the taxi should do) and you'll find some nice walks in the hills.

Zhouzhuang is a tiny water-village, in which you can walk around some old houses and canals. Jinxi is the same, but less touristy. We chanced upon Jinxi one day and can find little web-based resources about it -- it's on the way to Suzhou. Driving to Suzhou is easy -- you just have to get on the A11 to Nanjing, and at some point there will be a big sign saying "Suzhou". There will be some signs saying "Suzhou Outskirts" or "Suzhou City". Ignore the first signs that take you off the motorway/highway. Keep going until you reach the Suzhou toll-gate. The expressways are tolled in China, but they are inexpensive. Maybe a one hour journey will be 25 yuan.

To drive to Hangzhou, simply follow the Yan'an road as before, and look for the signs first to the A20, then to the A8 Shanghai-Hangzhou expressway -- easy.

If you do drive on the expressways -- beware. A friend of ours was once driving along (much further towards the southern regions) and suddenly saw a chunk of the road had been cut out. Yes, cut out. He described it as a "grave for a truck", saying it had perfectly been cut, and was about three or four foot deep, with no warning signs. This is an isolated incident. A common one is that the vast majority of the drivers on the expressways are as unpredictable as they are in the town, and drive just as badly. At night time, people who are driving on their work time will speed down the hard-shoulder to get home as quickly as possible.

Other than that, enjoy your road trip -- just make sure your map is trusty and your mobile fully charged!

Related:
Car renting business hot in Shanghai (Chinastic)
Rob Gifford took a 3,000-mile, 14-day trek from Shanghai to Korgaz and makes an audio report of his experience (NPR)
A site with two tiny paragraphs on two day-trips (World 66)
A day-trip to Jinxi (My Green Head)
A day-trip to Suzhou (with a small rant-based prologue) (My Green Head)
A day-trip to Hangzhou (My Green Head)

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