Chinese New Year is a time to welcome in the new year, in hopes that happiness and fortune (that we've hopefully already encouraged into our lives) will stay and prosper. There are a lot of traditions surrounding this time, and many taboos, or things we shouldn't do. Most of these are ways of discouraging bad stuff from lingering (or coming back) to us.
I find that a lot of New Year traditions (both Chinese and Western) are based on the idea that New Year Day (or the days of Chinese New Year, if you honor the full Spring festival) is representative of the whole year, so the stuff you do (and don't do) on New Year's day will follow you for the rest of the year. (note: some of the restrictions are based on particular days of the full Spring festival, so some restrictions may be for a slightly longer time than just the day, or for a different day than the actual day of New year)
With Chinese traditions, strictures are often included for homonyms: words that sound like other words. This is a very common idea, that words that sound similar are linked, so acting out one of the words will bring in the energy of the linked word.
Speaking of words, one of the big warnings is to not say unlucky words. This basically means don't talk about stuff you don't want to happen in the next year. Typical words to be avoided are: death, illness, pain, suffering, poverty, loss, killing, etc. This is one of the places where linked words are also avoided, so the number four is avoided (as it sounds similar to the word for death). However, you can use euphemisms, so instead of talking about someone who has died, you might refer to them as having moved on or being in a better place.
On a similar vein, you should avoid fighting and crying. Again, the idea is if you do these things, your year will be filled with fighting and loss. Children are often given free reign (and not punished for things), to avoid scolding them (and thus them being scolded all year). This also often leads to acts of peacekeeping, so neighbors might come by and make amends for any issues that came up over the past year, so that the new year will be full of peace.
Sweeping shouldn't be done, nor should you take out the trash, as you will be sweeping away your wealth or throwing out your good fortune. If you do need to sweep, you should sweep from the outside in, so that you are collecting your money instead of sweeping it away. It is seen as unlucky to wash your hair, as hair and wealth are homonyms, so again you would be washing away your wealth. For similar reasons, you shouldn't cut your hair.
There are several traditions around clothes. It is best to wear new clothes, for a fresh start to the new year, but you should absolutely avoid wearing damaged clothing (you don't want to wear rags for the whole year!). Black and white clothing are traditionally associated with somber occasions (like funerals), so they should be avoided.
You shouldn't wash your clothes either, as not only are you (again) washing away your wealth, but the water god is honored (on the first days of the new year), so to wash on his sacred days is seen as disrespectful.
Sharp objects shouldn't be used, which includes scissors, knives and needles. Using sharp objects are often seen as encouraging quarrels and conflicts in the new year. Sewing, in particular, is not to be done, partially as a break (mostly for the women), but also as it is felt that if you sew on the new year, the coming year will be full of work.
You should avoid breaking things, but if you do break something, you can wrap it in red paper/cloth or say a blessing over it (to turn away the bad luck, a common blessing uses the world year, which sounds like broken in Chinese to 'turn around' the energy of breaking stuff)
Porridge shouldn't be eaten for breakfast (because it's a poor person's food, and you don't want to start the year off 'poor'). A vegetarian breakfast can be seen as a way of demonstrating that you are in good health and don't need to eat extra meat. Some feel meat shouldn't be eaten at breakfast as a way of respecting Buddhist gods (who are against killing animals). Killing (even of animals), of course, should also be avoided.
I think it's interesting that you shouldn't wake people up, as that will lead to them being urged to work extra all year. But you also shouldn't take a nap, or you will be lazy (and it's expected that you will have visitors, and it's rude to be napping when people come to call).
One should avoid visiting hospitals, as this is believed to bring more illness in the coming year. Many people also avoid taking medicine, however this is often reserved for only minor issues (like maybe you don't pop a painkiller for a sore shoulder), and of course life-saving medications and emergency situations are exceptions.
You shouldn't lend or borrow money, as this will encourage debt. Gifts are often a part of Chinese New Year, and if you gift money, it should be in even numbers, especially lucky numbers (like doubles). Unlucky numbers (like four, mentioned earlier) should be avoided. Also gifts with unlucky associations should be avoided (like watches/clocks which are a homophone of paying last respects)
This is just some of the more common, or well known taboos, but you can see the trends here, and it mostly boils down to: do stuff that brings luck and avoid things that bring bad energy. And those are just good tips to follow, in just about any situation!