I use KeePass for this, as it is open source and cross platform. If you want cloud you can place the password database on one of the many cloud storage services. It correctly handles multiple devices saving to a common database file, so saving new credentials from both your phone and your PC will merge the two correctly when saved.

Edit: You can also find all kinds of security fuckups when using a tool like this. Biggest one I ran into was the wife's HSA provider website. They allowed us to create a password 64 characters long, so we did. Turns out under the hood it was chopping it to the first 16, but the login page takes all 64 then declares no match. Failed again with a new 32 char password but worked with 16. So tried putting the original 64 one back and only typing first 16 chars. that worked. Holy shit batman just only take the first 16 chars for password input if you're going to chop their password elsewhere anyway!

I use KeePass for this, as it is open source and cross platform. If you want cloud you can place the password database on one of the many cloud storage services. It correctly handles multiple devices saving to a common database file, so saving new credentials from both your phone and your PC will merge the two correctly when saved.