The only sane approach is a password manager. I use KeePass. This will generate new passwords of whatever length or alphabet you want, and keep them in a database. The database is encrypted, so you can back it up to non-secure locations. I use DropBox as an off-site backup and as a way of replicating the database to a variety of devices - desktop, tablet, phone. I have to remember the KeePass master password (which is a long, nonsensical phrase), and the DropBox password.

You can get add-ins for browsers that will attempt to recognise web pages and enter the correct password for you. I found they weren't reliable enough, and they also mean having the database open all the time you are browsing, so now I just copy and past between KeePass and the browser as needed. No-one shares my machine so I don't mind leaving websites logged in, so I don't need passwords every time.

I actually keep two password databases. The second one makes low security passwords more convenient. It has an easier master password that I can type quickly, and I don't mind leaving it open for extended periods. I use it for websites that don't have much at stake, especially forums.

There are several other password managers. I like KeePass because it is open source, and stores its password database locally. Some others store their database online, which means you can get to it from any device, but I think means you have to trust them more. Whatever you use, it should give you strong passwords that you don't need to memorise, and it avoids you ever having to reuse passwords. Just make sure you don't lose that password database or forget the master password.