New carpets are a well-known cause of environmental illness.
Synthetic carpets release poisonous gassed for quite a while after they have been installed.
If you are considering the installation of new carpets, I advise against it.
If you are stuck with new carpets, here are some things you can try to reduce the damage.
I suggest you ventilate your home as much as possible – keep windows open as much as you can (weather permitting). If you have central heating/air conditioning, the system should have a fresh air intake. Make sure it is open to the max and leave the fan running 24/7.
If you have any extraction fans (eg from the bathroom or kitchen) that exhaust to the outside, leave them on 24/7. Note that many kitchen fans do not vent to the outside (crazy I know, but common – go figure) – so check this before bothering to leave it on all the time.
Get an activated carbon filter for your home. Hardware stores often sell HEPA filters with integrated activated carbon filters. The HEPA filter does nothing against the gasses, it is the carbon filter you are looking for.
Sarah has left a comment suggesting that a photocatalytic air purifier is helpful in improving air quality (available from Amazon, follow link in this sentence).
Important Note: I don’t want to scare you but you have to know: new carpets can seriously destroy your health.