The browser wars are ongoing, but there is a clear leader on Linux at this time. This winner is Chromium. Chromium is the open source base that Chrome is built from, but it doesn't have any of the proprietary parts or unknown data reporting built into Chrome.
How to install Chromium on Ubuntu
The easiest way to install it is to add the PPA to your repositories. You can follow the instructions on the PPA directly, or follow the instructions below, assuming you are running Karmic or later.
sudo add-apt-repository ppa:chromium-daily
sudo apt-get update sudo apt-get install chromium-browser chromium-codecs-ffmpeg-nonfree
How to Get Started
Either run chromium-browser directly, or choose it from your menu. Chromium supports extensions, as well as bookmark sync that uses your existing google account. Find these things in the preferences menu, which can be opened by clicking on the wrench icon in the upper right hand corner.
Youtube HTML5 Videos
Until youtube offers full support for HTML5 rather than Flash for videos, you can install the following extensions which swaps out the page components for you: YouTube HTML5-ifier which is in development by Mark Renouf and myself.