(Posted in 2 parts):
It depends upon exactly who you're trying to keep your activities secret from. The latest versions of most web browsers have a 'private browsing' option which prevents the browser from recording the websites visited, so that others who use the same computer can't see where the user has been looking.
However, when your view a website that site needs to know where to send the data to. It does so by recognising your IP address, which is assigned by your ISP. Anyone with access to the log files of a website you've visited can see what your IP address was at the time of your visit. For most users that's not a problem but terrorists and people downloading child porn (for example) can be identified if the relevant authorities obtain a warrant to get the ISP to identify who that IP address was allocated to at the time of the download.
It's possible to hide your IP address by using an anonymous proxy. There are various ways to do so. One method is simply to Google to find the relevant details for a possible proxy and to enter them into your browser's settings. (I sometimes send email, using a web-based account, through a Brazilian anonymous proxy server, so that subscriber's to my posts don't know which country I'm in. Even so, if I was to do something illegal -which I don't - it might be possible for the 'authorities' to trace me by getting hold of the log file of the proxy server).