As with most Internet security concepts, you need to determine what you are trying to protect against. Anonymity on the internet is a relative concept. If you want reasonable anonymity, use TOR or a paid VPN/Proxy service. If you go the former, be sure you read and understand the terms of service. In your browser, use add-ons like noscript, ghostery, adblock plus, and flashblock to remove and block most of the tracking cookies and systems. Be mindful of where you visit and especially of applications like Java and Flash which work as applications on your system and have access to more data than just your browser.
While privacy is a right, if you are doing things that are illegal, you can assume that you are not anonymous and you will get caught, period.
WRT your other question, yes, the easiest way would be to use a program like VNC.