I put them on my online email inbox so that I can copy and paste them as I need them.
Ahh! So your email provider has your password. Shit. If you're going to put your password on the cloud, at least client-side encrypt it.
Sounds like you need to start using a password manager, such as LastPass.
Secondly, these passwords are randomly generated characters. So I can’t ever remember them
Most passwords you can remember are not secure enough. Ideally, you switch to having one master password for your password manager that is very secure AND you remember. The rest of your passwords are long strings of random characters/numbers and are remembered by the password manager.
I have to go try LastPass. Are the passwords stored locally or on a cloud? Meaning if I change devices, do I have to remember to put back everything or I just need to remember the LastPass's password?
Passwords are backed up to the cloud, but in an encrypted way. Therefore your passwords will sync across all or your devices (which I agree is essential convenience). However, your master password never leaves your computer. A good litmus test is that if you loose your master password, there is no way for LastPass to reset it. They may have a less secure option in which they can reset your password (and hence have access).
My only gripe with LastPass is that it's not fully open source, which is a BIG GRIPE when it comes to security critical software. But I'm not aware of an open source option that provides syncing across devices.
Also never use the same password on multiple sites. The password manager helps here as well.