We brought our baby over at 5 weeks for hi mum's maternity leave and then at around9 months old moved over until he was 2. From that experience I would bring the following:
1) Umbrella folding Stroller/Pram if you plan to take taxis, otherwise you can get the flat folding ones if you have your own car and will use it all the time. Maclaren comes to mind as our lasted well into his 4th year even after 2 yrs in Shanghai, though certainly international brands like Peg Perego etc will also do fine. Just make sure its rugged as the roads, sidewalks etc here really beat up the strollers/prams
2) Car seat- As noted by someone else, bring one from a trusted brand but make sue it has all the needed attachments to install in whatever local car you will be getting. Not all local cars may have LATCH attachments, so try and check first.
3) Pediatric Med, ones that you are used to and can read the directions on. Our baby had digestive issues once and we had to figure out how to administer the stuff that the local hospitals administered. Turned it was probiotics and some sort of clay powder and it worked fantastic, but trying to figure it out is a challenge.
4) If you are not breast feeding anymore, then by all means load up on your fav brand of formula and baby foods, especially if you child is a picky eater. Nothing worst that trying to transition a child's food to something you are not familiar with yet. Nonetheless, you will need to transition but just make sure you have enough to make a smooth transition.
5) Shower filter- or something that can reduce the chlorine in the water that you bathe baby in. Depending upon where you end up living, the tap water may be overloaded with chlorine as an attempt to disinfect it for household use.
Things you can get cheaply in SH include:
Diapers- try many brands and find one that suits you. No point taking up all that space in shipping container. Besides, if you get an ayi to take care of baby, they will probably potty train them much earlier than many of us think is possible. (amazing savings on diaper costs

)
Toys- while one needs to be cautious b/c of possible contamination and such, there really are many good choices in the local market.
Clothing- So much selection not even funny. Visit Ni Hong Children's Market and similar places to best deals . And Qi pu Lu market has floors dedicated to kids but it is not for the faint of heart. (See other theads for details)
Hope this helps!