The browser is definitely another thing to secure, I don't know that cookies are a danger specifically but ads and tracking services as well as insecure HTTP connections are all things to watch out for.
My setup for web browsing is latest Firefox running
uBlock Origin and
HTTPS Everywhere, as far as I have seen this addresses the majority of security concerns on the web
Taking it one step further you could use things like
NoScript and
LibreJS but they are much more aggressive and tend to neuter the functionality of a lot of webpages, so I've never been able to stand keeping them enabled for very long
There's also
Tor of course for near-complete anonymity but unless you are pursuing illegal content or attempting to subvert censorship I wouldn't see much reason to use it. You'll find the odd service that exposes itself over Tor (The Pirate Bay is a well-known example) so it might be useful to learn how to set up & operate it, maybe it'll come in handy for a very extreme edge case someday.