After I install the bitcoin client, what are the risks of browsing the Internet? Can my wallet be stolen?
Are there any known viruses/websites etc....?
Money can now be stored in a single file inside your hard drive. This is a natural target for criminals so yes, there are virus that steal bitcoins.
If you trust a third party (like an ewallet) to hold your coins, you will not have to worry about wallet stealing virus nor about wallet security. Instead, you should protect your login credentials and/or use a two-factor authentication mechanism. Treat your e-wallet account the same way you treat your e-banking account.
I don't know of any dangerous websites, but there was a report of a bitcoin trojan by Symantec on June 2011.
Yes they do exist, your best bet is to create a secure wallet . That link goes to another question about how to create a secure wallet.
Yes, and they are more dangerous than they first appear
Consider the situation that a wallet stealing tojan infiltrates your computer/smartphone. All it has to do is make a copy of your wallet.dat file (or equivalent) and send that copy back to base. It can then safely delete itself since it has completed it's mission.
Now comes the dangerous part.
Over the course of time confidence in Bitcoin grows and you make the fatal error of not securing your wallet and start to keep larger and larger sums of Bitcoins in it. Years may pass. Eventually, the person who now has a duplicate of your wallet (a black market in trading these duplicates may arise) may decide to cash out. All you see is that a very large anonymous transfer suddenly occurs out of your wallet and into theirs. You'll be completely powerless to prevent or recover these losses.
So, make a secure wallet.