I think what you're seeing are pop-up ads that try and sucker people into clicking on them with phrases like "Your internet could be going much faster -- click here to remedy" or "Your computer appears to be infected by blah-blah -- click here to clean". Those are not legitimate warnings -- besides, your web browser will not be the application to notify you of infection if your computer is ever infected, so don't trust web browser messages.
If you can describe "weird things" in detail, we might be able to help you sort things out. As of right now, though, it's definite that your computer is not infected with spyware, malware or viruses -- your problems are coming from something else.
A good thing to try when experiencing weird things is to simply create another temporary user account using the "Accounts" pane of the System Preferences and see if the weird things persist when logged in under that new account. If they do, it's most likely a system-wise problem -- if not, then it's most likely something to do with your previous user account.