I use Password Safe myself; there are plenty of similar tools out there.
Most of my passwords are randomly generated. I use, for example, 12 or more random lowercase letters when I can (easier to type on mobile devices), though some sites require a mix of upper and lower case letters, digits and punctuation.
I never use the same password on two or more sites. It's easy enough to generate and save a new password that there's no advantage in doing so.
I don't even remember the vast majority of my passwords, because I don't need to.
This does mean that I need access to my password database, and that database is potentially a single point of failure, but it's worth it. The database itself is, of course, encrypted (with a password that I have memorized); I keep it on a USB thumb drive and keep backup copies in safe places.