A "traditional" Chinese breakfast varies greatly from region to region. Congee (粥 - zhōu) is a type of porridge/gruel made of rice, and is generally a Chinese breakfast mainstay. It usually comes with the option of toppings, such as eggs prepared in a variety of ways, onions, fried garlic, sesame oil, chili oil, meatballs, fish balls, steamed fish, pickled lettuce, fermented soybean cake, salted black beans, pork or beef jerky, fried bread, beef tripe, etc.
However, those looking for something that sounds a little more palatable to start the day also have many breakfast options in Shanghai.
Western breakfasts and weekend brunch
For western-style breakfasts (eggs, bacon, sausages, etc.) check out this thread about western breakfasts and brunches in Puxi.
Weekend brunch is fast becoming something of a Shanghai institution - whether it's simply getting a fry-up or going to a five star hotel for a buffet breakfast with freeflow champagne. A recent article on TimeOut Shanghai features the best brunch deals in town, from the sub-100 RMB price range to the various luxury hotel options that can set you back as much as around 800 RMB (plus a 15% service charge, of course).
No account of breakfast - or rather, morning snacks - in Shanghai is complete without mentioning two Shanghai specialities: steamed soup dumplings (小笼包 - xiǎolóngbāo), and pan-fried dumplings (生煎包 - shēngjiānbāo). Both of these are widely available at local eateries, and are both basically a meatball wrapped in dough.
Xiǎolóngbāo come in a variety of flavours - most commonly pork, but also other meats, vegetables, or crab. Be warned, however - to eat them, pick one up carefully so you don't pierce the skin, then bite a hole in the top. Blow into the hole to cool down the soup then suck it out, then eat the rest of the dumpling.
Xiǎolóngbāo shouldn't set you back more than 5-10 RMB for a steamer, and there are usually 8 xiǎolóngbāo per steamer.
Shēngjiānbāo are also usually filled with pork, but you will occasionally see chicken, or pork mixed with crab or prawns. Again, as they are usually served fresh from the steamer/pan, not biting a hole and sucking out the soup first is a surefire way to burn your mouth and ruin your enjoyment of the meal.
Again, you should not be paying more than 5-10 RMB for a serving of shēngjiānbāo, which usually come in servings of 4.
Street food breakfasts
To see what your options are in terms of Shanghainese street food breakfasts, check out this article, which outlines some of the more commonly seen street food breakfasts and snacks around the city. Not only are many of them delicious, but they are also very cheap, usually ranging from 2 to 5 RMB in price. Here are some of the more popular options:
Another common breakfast item is the humble baozi (包子 - bāozi), which is simply a steamed bun with meat or vegetables inside. They generally cost 1-2 RMB each and come in a wide variety of different fillings, though the default is pork.
A jiānbing (also known as a 蛋饼 - dànbǐng) is a Chinese egg pancake. The base pancake is thin, not unlike a crêpe. Jiānbing vendors have a variety of different recipes, but the standard jiānbing is a pancake with an egg spread out and cooked on it. Vegetables are added (some fresh, some preserved), then it is folded over on itself. At this point usually 2 sauces are added: first a sweet, hoisin-based sauce is spread across, followed by a light dabbing of chili paste. Then, as the pièce de résistance, a deep fried wonton skin is placed on top and the pancake is folded into a tube of deliciousness.
Pro tip: when ordering a jiānbing from a street seller, always ask for 2 eggs rather than 1. This usually will cost an extra RMB, but greatly enhances the taste.