Dumpling soup is probably easiest to make. It's harder to replicate dim sum so that it is close to authentic, so I suggest going to an authentic Chinese restaurant for that. It's what a lot of Chinese people would do for Chinese New Year anyway, if they celebrate it.

Here's a lovely Shanghainese menu, quite different from the usual Chinese fare, more homestyle cooking:

http://mimithorisson.com/2013/01/28/hong-shao-pork-fava-bean-fritters-tea-eggs-2/#comments

Here's a pretty simple recipe for nian gao (mochi cake), a traditional dessert eaten at Chinese New Year: http://the-cooking-of-joy.blogspot.com/2009/01/nian-gao.html

You MUST MUST MUST have dumplings and noodles for Chinese New Year. If you have to make fresh noodles and dumplings, awesome! But if you don't, I like to use the store bought dumplings from Asian market (I poo poo the ones from American markets and Costco) and I order in lo mein from my favorite Chinese restaurant... :o)

Serve a whole fish with the head and tail still attached. I don't know what it symbolizes but it did make for a nice presentation. Or just go out for dinner at a dim sum place. A lot of places will hire dragon dancers. Bring some small bills to give them (they will go into red envelopes) and the restaurant will often be collecting for some charity.

Aaah, I love Chinese New Year, it's my favourite holiday of the year. I thought that Joyosity's link for nian gao was super interesting, as traditional nian gao (that I grew up with) is usually steamed and not baked!

Steamed chicken is a pretty popular dish at my grandma's house. Marinate a whole chicken (around 2kg) for at least two hours or overnight with salt. You can use smaller cuts if you're not cooking for a whole family. Steam the chicken over boiling water for half an hour. We then use the er.... chicken juice? that flows out onto the plate when cooking the rice. Oooh! Throw some ginger slices into the rice for the fragrance. We eat it with a ginger scallion sauce. Yummy!

Potstickers or wontons are easy to make, especially with store bought wrappers. I make extra to freeze and guests are always happy to have the frozen "party favors" to take home. Not a traditional New Year's dish, but I've been making lots of Hainanese chicken for casual get togethers (Steamy Kitchen has a good recipe) as well as marinated cucumber salad.

try http://7th-taste.com/2011/01/04/korean-new-years-dumpling-soup-with-rice-cakes/