For a long time I just let my browser (Firefox) remember them for me, but installed an add-on that let me view and edit the account and password information. It allowed me to add new entries whether the website existed or not, so I could make up some dummy name for account information that didn't have an associated website. That obviously wasn't a cross-platform solution, and I've subsequently realized that:

1. I was only using perhaps 10-12 unique passwords across all of the web sites I frequent (see below)
2. I needed something that worked on my laptop, tablet and phone
3. A way to manage other, non-password textual data would be very helpful

It appears my requirements aren't all that different from the OP's...

When I installed LastPass, one of the first things it did was read the Firefox password database and perform a security assessment. That was a real eye-opener, revealing just how often I was reusing the same passwords, and that the level of complexity was considerably lower than I had assumed.

so I took the time to put LastPass to work, auto-generating much stronger passwords and updating the most critical websites (along with the majority of rest). The password assessment report now shows an acceptably high value.

I've also implemented 2FA wherever it's supported, so I'm much more comfortable that I've done everything reasonable and prudent to safeguard my on-line access.

And I'm seriously suggesting you think a bit longer about whether a Google Drive hosted, password-protected spreadsheet is really safe enough.