No, it is not safe to assume that you have anonymity.
In general, anonymity exists in a spectrum (i.e., anonymity is not a binary attribute). Who do you want to hide your activity from?
If you operate completely openly, then you can have cookies enabled on any web browser and use non-encrypted channels.
If you wish to hide your information from potential eavesdroppers, then you can use a cookie-enabled web browser and encrypted transmission.
If you wish to hide the origin of your traffic from the person who receives your communication, this is slightly more difficult, but you techniques similar to spoofing will work well (i.e. Tor, where the traffic appears to originate from a different server than you are actually using).
If you wish to hide your traffic from all the servers you access, this is pretty much impossible, as your local ISP and local network administrator will certainly be able to see your traffic, even if they cannot read its contents.
If someone, like a government agency, really wants to track you down, they can, even if you use Tor and openVPN.
Your best protection is use the Internet in ways that are above reproach, such that complete anonymity is not necessary.