A little paranoia is a good thing!
Google already keep your search requests and have access to all your emails if you use gmail. Using their browser just adds to the information they can hold on you.
Firefox will delete all your cookies and search history if you require it when you close it down.
And don't forget that your ISP can use deep packet inspection to have the complete picture of your surfing habits. There isn't a great deal of privacy on the internet any more.