It all depends on what you mean by "fingerprint".
There's the fingerprint of the Internet services your machine provides (which port, which software, which version, if available), if it's connected to the Internet (remember, if you're connected to the Internet, the Internet is connected to you). You can minimize the size of this fingerprint by being behind a well-administered firewall, reducing the number of services your machine runs, running services on non-standard ports, editing version strings, etc, etc.
Then there's the fingerprint of the IP address your ISP has assigned you. TOR (see previous answer) can help with that, but with a sufficiently detailed examination, one might be able to see a 47.853GB download of a forbidden document from some evil website to a TOR node, AND one might be able to see a 47.853GB encrypted file from some TOR nodes to your IP, and draw conclusions.
Then there's the fingerprint your browser presents to the sites you visit (via TOR or not). The PanoptiClick project of the Electronic Frontier Foundation https://www.eff.org/
to see how unique your browser configuration and add-ons are.
Then there are actual fingerprints - wear thick gloves.
First, consider what you want to secure, then from whom (well, really, do a Threat and Risk Assessment), and figure out how much effort/time/money it will take to secure it. Is your "fingerprint" your machine provides to snooping parties on the internet, really the highest risk?