While my old answer below is still my recommendation (I updated it), Firefox has a new Testpilot Feature that does exactly what you want: It allows you to use internally a different browser for different webpages if you want to. You can also automate this process and therefore make less mistakes. This feature will probably included by default in a future Firefox release this year (2017). So, give it a try, use Firefox, browse to https://testpilot.firefox.com/experiments/containers and follow the instructions.
old, but not outdated answer
Using two different browsers could partially help. But I remember reading about non-Facebook users getting tracked by Facebook on third-party sites. They probably just wait for you to log in. It's better than nothing, but just do a mistake once and they got you ;)
Once they have the connection between your second browser and your account, they can always see who you are based on some new HTML5 canvas tracking techniques for example (no cookies needed). I don't know whether they use this, but basically it's possible. Also it's a lot of work and distracts your workflow. I wouldn't do it, but maybe someone can rate this on a more professional level.
I use (and recommend) a combination of Firefox addons to protect privacy. Note that addons for your browser in general can make browsers slower, e.g. on older computers. Especially if you add too many lists in ad blockers. I recommend one or two good lists.
uBlock Origin: A good ad blocker will block most ads and tracking stuff based on lists. I do not recommend AdBlockPlus, because they have a strange business model, and they had some legal arguments at least here in Germany. I didn't find any good English source, but fact is they have a conflict of interests, because they are working with and getting money from companies to whitelist their ads under some criteria (they call it "Acceptable Ads"). So better ignore AdBlockPlus, if your concern is about your privacy.
Download: uBlock Origin
Privacy Badger: This is a work-in-process plugin that does not work with lists, but with algorithms. It's developed by the Electronic Frontier Foundation (EFF) to protect privacy. Still, it's in development and (probably) not yet perfect, so I use it together with AdBlockEdge. There is no configuration required.
Download: Privacy Badger
HTTPS Everywhere: This enables HTTPS when possible, even if some sites do not support it by default or in a broken way (that is, you are on https, but as soon you click on a hyperlink, it's gone). It is also developed by the EFF. There is no configuration required.
Download: HTTPS Everywhere
A few words about Privacy Badger
Privacy Badger rates every connection to "green", "yellow" and "red". Red means blocked, green means not blocked. Yellow means that only cookies from this connection are not accepted. I recommend to stay with the rating of privacy badger as long as everything works. You can easily change this though through a slider for every connection.
Screenshot on heise.de
This means that it will do compromises (yellow), trying not to break the sites functionality.
Solutions for Website Operators
Of course it is disappointing to see that your favorite sites are not really caring about your privacy. You could ask them to improve, e.g. by using the "Sharriff", which generates new sharing buttons without tracking support. They do not use the default sharing buttons, but simply generate a URL to your social network. So you only get tracked when you click on a sharriff button. Basically, it's like manually sharing the link and it looks pretty normal to the user. It's available on GitHub and was initiated by heise.de, a big German technology magazine publisher.