It may not be possible to completely scrub yourself from the Internet, but you could lower your profile somewhat if you try.
First, understand where Google (or Bing or whatever) is pulling up your data. Then, take steps to remove yourself from those sources.
1. Data Brokers
These are the scariest culprits. When you Google yourself you might see your name, city and age popping up on sites like MyLife, BeenVerified and Instant Checkmate, among others. You click and see some information about yourself with a giant button promising more.
A few years ago I went through the tedious process of requesting these sites remove my name from their public search listings. But they still have all my information for sale. Just now I searched my name on Whitepages and found I appear in the search results again, but this time as a "premium" result.
It's enough to unnerve any of us who values our privacy. But there's really not much we can do about it. Data brokering was around long before the Internet and they get information from numerous public and private sources.
What you can do is minimize what appears, which might keep Google from showing it in a search result. You can do this by going to the sites one by one, like I did, and go through their "removal" process, which only removes a few telling details from the public search results. Or you could pay a "privacy protection service" to do it for $100+ per year.
2. Social Media
If you don't want your Facebook, LinkedIn, or other social media accounts to appear in Google searches you have two options:
1. Figure out how the privacy controls work on those sites.
2. Delete your accounts and don't join those sites again.
3. Employers and other Associations
Maybe there's a profile about you on your company website. Or you're mentioned in the online newsletter of your church, baseball club, or other association. If that makes you uncomfortable, talk with the website administrators about removing your name.
4. Old Embarrassing Stuff
Opinionated forum posts, silly photos of you on your friend's old blog, that time somebody interviewed you for a news article, anything that shows up on a website you associated with in the past might still haunt you in search results. Contact the site webmaster or administrator and ask if they can take it down.
That's really the best you can do to remove yourself from search results, or at least show up in fewer results. If you're concerned about your online reputation, you might also consider an opposing strategy: get yourself in search results above the links you don't want people to see.