Do not use the same password on different websites.
Do not use Tor browser. Using a Tor browser puts you in the risk getting your bitcoins stolen.
Do not get involved in transactions outside the LocalBitcoins.
Do not use the website from a shared computers or devices, like ones in public internet cafes, as they may have keyloggers installed to steal your user credentials.
When logging in to the website, read the browser address bar and check that you are logging into localbitcoins.com and not a phishing domain. Make sure the spelling is localbitcoins.com exactly, as the phishers, especially email phishers, often register domain names resembling localbitcoins.com domain name. (ie: localbiRcoins.com).
If possible, when accessing Bitcoin wallets, do this from a dedicated computer you have reserved for financial tasks only. Do not use this computer for other tasks.
Do not install third party software, pirated software or browser addons you cannot trust 100%. This greatly reduces the risk of getting Bitcoin stealing malware infection on the computer.