Also from experience, most sites will implement it in every wrong way possible.
For example, all the major sites that allow the total of 1 active TotP authenticator app - trying to add one forces to delete the other. Which is fine while you have only one phone and aren't in the process of switching to another one.
With foresight you can get around this since you can reuse the TOTP seed values.
The annoying thing is so many services don't even support TOTP. They either want their own proprietary app, still insist on SMS, some of them even try to get you to use voice prints!