This is a very complex topic. You might start with Wikipedia and work your way through the various types of threats out there. Adware, Malware, Viruses, Denial of Service, Rootkits, Botnets, Worms, Spyware to name a few.
If you stick to mainstream websites you are less likely to encounter malware and viruses. It is people using search engines and venturing to lesser traveled parts of the web that are most susceptible to infection. Less traffic, less people reporting problems, and the lesser likelihood of there being a full time staff to handle such issues.
I would say the most common type of malware will come in the form of advertisements. Depending on the source of the advertisements nasty people can sneak in malicious code every...