I've tried a handful of of different browsers over the years, mostly stuff in the Debian/Ubuntu/etc etc repositories but I've always fell back to Mozilla Firefox (or Icecat if you're using Debian GNU/Linux ;). This is mainly because of a few choice addons. There may be ways to do some of this without them and in fact I edited my network IP block settings from a list somebody compiled to simply not even load known advertisements and other bad sites. I did this on WinXP & Linux and it worked great for a while. The following is pretty much a must have list of modifications if you want a fairly secure experience.
But that isn't all you have to do! I always disable cookies from Third Party sites and I'm not sure if I've ever once had to allow these. The only cookies I really want are for logging into certain websites not unlike the one we're on right now. (Interesting how "now" is actually referring to two different times. For me it means the time of this writing but for you it's any later date when you're reading it.) You should also regularly clear cookies, history and all that good stuff. There's an option to delete everything or select file types automatically when you quit the web browser. There's a "private browsing mode" in some browsers. Personally right now I'm not using it but you can if you want.
I actually recommend using third party open source alternatives for most things regarding personal data because the built in ones are almost always going to be the most exploited because everybody with that browser is using them. Just look at Internet Exploder, I mean Explorer. Oh, by the way never use that for anything. I think the only reason to have it installed is 1) because I don't think it's possible to uninstall it (so keep it up to date) and 2) it's needed for M$ & Win updates. I recommend keeping it as locked down as possible and remove all shortcuts to it.
If you want to keep track of your previous session-those dozen tabs you had open before FF closed then check out Tab Mix Plus.
There are a bunch of other helpful or useful ones so check them out and experiment. Don't be afraid to mess with your computer (so long as you're important data is backed up somewhere other than that computer though you shouldn't have anything to worry about with web browser and their addons).
Some time in the future I'll list open source and privacy respecting alternatives to Google and other sites on the internet. For now I will mention: DuckDuckGo as a great web search that anonymously uses Google and other search engines to compile results.