You have a week in Shanghai, which is plenty of time to also do day trips outside the city.
First thing I'd do is to take an Untour food tour. They get rave reviews on Tripadvisor. It's a food tour of the city that's run by several expats who've sourced out all the best street food and little restaurants. Take the night tour:
https://untourfoodtours.com/ and do it early in the trip to discover the little restaurants you'd like to return later in the trip. Note: there may be one or two interesting food at one of the restaurants they'll take you to, like snakes, but you don't need to eat it! The vast majority of food will be perfectly edible and terrific. Book your tickets online in advance and they also sell good guide books they've written and which you can download and which are excellent for getting ideas for where to eat (quite cheap, I think it's $6).
I'm sure you have guidebooks and it's worth it to take a self-guided walking tour through the old parts of Shanghai (the old Chinese city). Atmospheric. I really enjoyed Yuyuan Garden. A great museum is Shanghai Musem.
For day trips, go to Suzhou (30 minutes on the bullet train) and visit the fantastic gardens. Hangzhou is further afield but is famous for its large lake and is very scenic. There are also a number of "water towns" scattered between Shanghai and Suzhou/Hangzhou which are old historic Chinese towns built along canals and can be atmospheric. It's a bit tricky getting there on your own but is feasible, or you can see if there are organised day trips (ask your hotel). Tongli is one of the bigger towns and it's near Suzhou (note, it's really not practical to try to see the famous gardens of Suzhou and Tongli on the same day). Xiteng is another nice one, small but not as crowded as the others. Zhujiajiao was where Mission Impossible was filmed but can be crowded (hit or miss).
Another good restaurant that comes to mind is Shanghai South Beauty, which is in Pudong on the top floor of a mall right by a metro station (the first one as you cross over to Pudong from the Bund side). They have fantastic views of the Bund and the food is quite good. Can check it out yourself on tripadvisor.
Do try the custard cups (imports from Hong Kong). These are little tarts, very flaky and filled with a baked custard filling. Lillian's is a local chain/brand and you'll see them around.
In planning a trip I find it useful to review the wikitravel page, and compare it to guidebooks, random selection of travel blogs and the sites themselves on tripadvisor. You'll soon get a feel of what's appealing and what probably isn't. Opinions vary, of course, but I've developed an intuition of which tripadvisor reviews to trust and not.