The most basic thing that everyone can do is also the most critical, and that is hide your IP address traffic. This is the cornerstone of any situation that requires deniability. The advice about using TOR is great, but I frequently take this a step or 2 further.
You can subscribe to high-speed VPN (Virtual Private Networking) services overseas for less than $20/month. I particularly look for outfits located in countries with a track record for resisting/refusing US law enforcement requests for electronic forensics. Croatia and Sweden rank high on my list, with Malaysia a close 3rd.
You can log-in to your VPN server overseas, and add a whole new level of IP spoofing to the security mix. Basically, you would first open TOR on your local machine to begin masking your activity. Then log-in to your VPN server, which is basically like opening a remote desktop environment. Your VPN has its own operating system, apps, browser, etc. Once you are logged in, then open TOR on your VPN server. You are now officially a ghost... as long as you obey the TOR browser and refrain from opening public links.
I actually have access to 3 VPN outlets, one of which provides for a Port 80 256 bit encrypted data tunnel, and I will occasionally connect to 2 and sometimes all 3, each using a TOR route buffer... But I only do this when I am meeting with the other Illuminati members to discuss world domination.
The advantage of doing this is that you can go almost anywhere and do almost everything online while maintaining the all important deniability.