Timeline pages for Facebook users who aren't your friends are still visible on the social network, though the information you will be able to see depends on the privacy settings of the Timeline in question. For most profiles you'll be able to see a cover photo, a profile picture, a list of mutual friends and some basic biographical information. You can find Timelines for non-friends through Facebook's own search tool or a general Web search.
Log into Facebook and type out the name of the user you wish to look for in the search box at the top. Click on the correct name when it appears or follow the "See more results" link to run a wider search.
Click on an existing friend's name then follow the "Friends" link to find users who are friends with someone you're already connected to on Facebook. Click inside the "Search Friends" box to look for the relevant person in your friend's friend list.
Run a Web query through your search engine of choice if you still cannot find the relevant user on Facebook, appending the word "Facebook" to the end of your keywords. Click through on any matching results to view the Timeline.
Select "Add Friend" on any user's Timeline to make a connection with them. You may also have the option to send a message depending on the user's privacy settings.
If you are not friends with someone on Facebook, only content marked as "Public" or will be visible on that user's Timeline. Public content can include posts, likes, biographical information, employment history and the friends list. Every user's current cover photo and profile picture are always set as public. If you have a mutual friend with the user in question you'll be able to see content set as "Friends of friends" too.
Regarding your own Timeline, open the Account Settings page when logged into Facebook to set the visibility of content on your profile. You can set whether search engines are allowed to index your page from the Privacy tab and limit the visibility of your posts and other content from the Timeline and Tagging page. Open your Timeline and use the audience selector drop-down menus by each item to make particular pieces of content (such as your photos or your relationship status) public or private. Open the drop-down menu by the black gear symbol on your cover photo and choose "View as" to see your Timeline as it appears to Facebook users who aren't confirmed friends.