swannybc :
Are websites able to read cookies installed on your computer, even after you install and use a VPN?
Yes they are. All those facebook and twitter icons you see on websites? Those are used to track you. There are more insidious tracking mechanisms than cookies too. You should also be using adblock, since one of the non-cookie tracking methods is for ads to run a script which looks at your hardware config to uniquely identify you.
If you're concerned about privacy, make sure to install Ghostery. It'll block most of the tracking cookies you'll run across, as well as some of the non-cookie tracking methods. Firefox also has options to delete all cookies, browser cache files, and history. You can do this manually, or set it to do this every time you close the browser. Firefox and Chrome also have incognito/private browsing modes which do the same thing for that particular browser window (they're also a great way to sign into multiple gmail accounts at once if, say, you use one address for personal mail and another for work mail).
You can also run flashblock (flash is one of the non-cookie ways to track you), to prevent flash apps from running unless you explicitly click on them. There are cookie managers too which let you whitelist/blacklist cookies from certain sites (I won't recommend one because my favorite stopped working with recent Firefox builds). And the nuclear option is noscript, which will block all javascript unless you explicitly allow it. The last two take a lot of work allowing just the non-tracking sites - it took several months before my whitelist was enough for me to browse normally. Ghostery does pretty much the same thing with a blacklist. (Whitelist = everything blocked unless allowed. Blacklist = everything allowed unless blocked.)