It is truly upsetting to see how few people use password managers. I have witnessed people who always use the same password (and even tell me what it is), people who try to login to accounts but constantly can't remember which credentials they used, people who store all of their passwords on a text file on their desktop, people who use a password manager but store the master password on Discord, entire tech sectors in companies locked to LastPass, and so much more. One person even told me they were upset that websites wouldn't tell you password requirements after you create your account, and so they screenshot the requirements every time so they could remember which characters to add to their reused password.
Use a password manager. Whatever solution you think you can come up with is most likely not secure. Computers store a lot of temporary files in places you might not even know how to check, so don't just stick it in a text file. Use a properly made password manager, such as Bitwarden or KeePassXC. They're not going to steal your passwords. Store your master password in a safe place or use a passphrase that you can remember. Even using your browser's password storage is better than nothing. Don't reuse passwords, use long randomly generated ones.
It's free, it's convenient, it takes a few minutes to set up, and its a massive boost in security. No needing to remember passwords. No needing to come up with new passwords. No manually typing passwords. I know I'm preaching to the choir, but if even one of you decides to use a password manager after this then it's an easy win.
Please, don't wait. If you aren't using a password manager right now, take a few minutes. You'll thank yourself later.
TOTP is standardised by RFC 6238 so all TOTP clients must comply with the standard and therefore work equally well. Pick the one whose UI you like the most and is otherwise good enough for your use case and personal preferences. It's similar to arguments over CPU thermal paste—its presence or absence makes a much larger difference than the method of application.
You do, however, want to pick something that is free and open-source and also popular. Google Authenticator (closed source) definitely is a functional TOTP client but you have to trust that the Google engineers have done a good job building a secure app. Since it's Google, they probably have, but a principle in security is that you should not have to trust more people than absolutely necessary.