If you're not using any password manager, then start using one right now. It's safe by any reasonable measure, and in particular, far safer than not using one.
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KeePassX is a great free software tool that helps you manage your passwords. It's like a safe. You use one password (key) to get into the safe, and inside the safe are all your passwords. Because you can store strong passwords, and look them up instantly, you never have to use weak passwords again. You never have to reuse passwords again. KeePassX has a password generator so it can create long complex passwords for you, and remember them too.
You can even use the password safe to store other sensitive information like Social Security numbers, credit card numbers, insurance info, access codes, license numbers etc. The "safe" is encrypted on disk, so if it were 'lost' then nobody could gain access to any of the passwords inside. What's more, KeePassX is cross-platform so that you can run it on any computer or device that you have. You can put the 'safe' on a network drive, or synchronized folder (e.g. using git-annex) so that you can access the safe in real-time from multiple devices.
Unlike online 'services' that offer similar functionality, you don't need to depend on them staying in business, or possible security breaches, or their profit model carving out features that you have to pay for etc.
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It's generally safe (meaning Chrome or Firefox) to use a browser's built-in 'remember my password' feature if you control the device and the login to that device. If you don't know what I mean by controlling the device, then you probably should just use KeePassX alone.
If you use Google Chrome for a browser, signed in using your Google Account, you can access your stored passwords either in the browser 'settings' (the menu icon in the top right corner of the browser), or online at https://passwords.google.com/ Sometimes when you forget a password, it can be quickly retrieved from the browser. The Google password system called "Smart Lock" was improved in 2015. As a final note: please use two-factor authentication in Google. It's simple and effective.